Area of an inscribed triangle, Central Angle Theorem, • Central Angle Theorem blackpenredpen, math for fun, blackpenredpen.com/bprplive, / blackpenredpen , blackpenredpen@gmail.com
Here BPRP I thought I should let you know i found two other methods to solving this. You can check here: cdchester.co.uk/wp-content/uploads/2018/06/Area-of-an-Inscribed-Triangle-4-Methods.pdf
Because Math is the language of universe and nature !! İf you don't know English, You don't understant people speak english only ;) But You don't know Mats, You don't understant anything !!!
@@supertester23 RHS = 2sin(A)sin(B)(2sin(C)) (some manipulation of original RHS 4sin(A)sin(B)sin(C) where 2^2 = 4 = 2sin(C)(cos(A-B) - cos(A+B)) (product to sum, 2sin(A)sin(B) = cos(A-B) - cos(A+B)) = 2sin(C)cos((A-B)) - 2sin(C)cos((A+B)) (expansion) = sin(C+(A-B)) + sin(C - (A-B)) - sin(C+A+B) - sin(C-A-B) (product to sum once again) = sin(C + A - B) + sin(C - A + B) - sin(A + B + C) - sin(C - (A+B)) (expansion of input of sin()) = sin(pi - B - B) + sin(A + B + C - 2A) - sin(pi) - sin(C - (pi - C)) (using the fact A + B + C = pi) = sin(pi - 2B) + sin(pi - 2A) - 0 - sin(C - pi + C) (sin(pi) = 0 and using the fact A + B + C = pi and expansion) = sin(2B) + sin(2A) - sin(2C - pi) (sine is positive in the second quadrant) = sin(2A) + sin(2B) + sin(pi - 2C) (sine is an odd function) = sin(2A) + sin(2B) + sin(2C) (sin(pi-x) = sin(x)) = LHS Alternatively, use the fact that BPRP's method and the method Heliocentric used to find the area must give the same area as they are the same triangle... so (1/2)r^2[sin(2A) + Sin(2B) + sin(2C)] = 2r^2sin(A)sin(B)sin(C) which means sin(2A)+sin(2B)+sin(2C) = 4sin(A)sin(B)sin(C) of course, if A + B + C = 180 degrees since we are dealing with triangles.
@@supertester23 Since this formula using the double sines gives the area, we know it is equal to the area BPRP derived in the video. By cancelling r^2 and multiplying by 2 you arrange to get the equality OP derived, and we have this true only for A+B+C = 180 because this is only true when the angles are angles of a triangle, which sum to 180.
You can do this alternative way to make things faster : Step 1 : by using the law of sin. For any triangle, the area will be 0.5*b*c*sin(A) where A is the angle between side b and c Step 2 : by splitting the the isosceles triangle in two the base and the angle will be split in exact half. Simply 0.5*b = r*sin(B)
Wow, then, in this time we can use the law of sine, a/sin(A)=b/sin(B)=c/sin(C)=2r, now formula changes like S/abc=2r^2 sin(A)/a sin(B)/b sin(C)/c=2r^2 (1/2r)^3=1/4r, so we can know that S=abc/4r ! :)
I did the exact same thing you are going to do in the future, finding the largest inscribed triangle for any circumference, and it was a hell of an exercise! Good to see you started up just like I did a while back, and I'm very eager to see if you do anything better than I did, and I sincerely hope you destroy my proof so that I can learn from my mistakes! Keep making these awesome videos!
I am ur biggest fan. The way u solve questions....it’s amazing thanks for coming on CZcams. And for this video ....thanks. This would help me in many olympiads in my 10th grade
In a circle of radius 1 , for any inscribed triangle with sides length a , b , c We have : (a^4) + (b^4) + (c^4) - 2 * ( (a^2)*(b^2) + (a^2)*(c^2) + (b^2)*(c^2) ) + (a^2)*(b^2)*(c^2) = 0
Thanks a lot bro you really helped me with my math homework and also helped me widen my interest in mathematics and also additional mathematics which is like a subject in IGCSE cambridge. Thanks
To solve b, you could also break triangle OAC into 2 equal pieces and calculate the area of one of the pieces(little right triangle) 2 ways. You will get 1/4 bh = 1/2 rhsinB, which is b = 2rsinB, a bit easier than law of cosine.
Much quicker to do like this: It's known that abc=4R△ where a, b, and c are the side-lengths. -- (i) It's also known that a=2RsinA, b=2RsinB, and c=2RsinC Using these in (i), we get: (2RsinA)(2RsinB)(2RsinC)=4R△ Or, △=(2RsinA)(2RsinB)(2RsinC)/4R=2R²sinAsinBsinC
Hello Steve, I've been subscribed to your channel for a while and I really had to thank you for everything I learned here. I'm taking calculations for the first time and their videos are very useful, I really love them. Please keep doing it and keep that good mood! PS You are very close to reaching 100k subscribers ... I wonder what you will do to celebrate :') Sorry 4 bad english :P
I think this is a pretty cool derivation for an expression for an area of a triangle inscribed in a circle. I just want to make one note about it; when you use the Law of Cosines in the triangle with vertices A, C, and the center of the circle, you assume that angle 2B is an angle in a triangle. This assumption translates to an assumption that 0 < 2B < 180 degrees, which is equivalent to 0 < B < 90 degrees. Therefore, the steps you have used in the derivation are rigorous only for angle B being acute, and since you argue the same things can be done with angle C to find an expression for c with r in it, the derivation also is only rigorous for the assumption as well that angle C is acute. I am pretty sure the final formula works for any triangle inscribed in the circle, but I think that there are a couple of additional cases to the problem to consider for doing the derivation by finding the sides of the triangle ABC you found, angle B being right or obtuse while angle C is still acute. I think the geometry and trigonometry of the derivation probably are slightly different to prove the final formula for angle B as right or obtuse. I hope you find this useful.
Ruby Duby, you're not far away. Use the fact that A+B+C=pi with the sum to product identity for sine and sine. Combine and simplify, you get 4sinAsinBsinC. Substitute back in your original expression and that's 2r^2.sinAsinBsinC
nice :) On a side note, i think if we equate the two formulas, we should end up getting a nice little property between angles. 2sinAsinBsinC = sin2A + sin2B + sin2C Itd be cool if this actually worked for any polygon besides triangle.
Skip the law of cosines. To calculate b, drop a perpendicular from the center (D) to chord b. Because ADC is isocelese, CAD = ADC. Right angles at crossing, imply angle on each side of perpendicular are equal and therefore b. Then opposite side is clearly 2 sin b. Same for c.
Construct a radius to each corner. The angle 'opposite' A (angle constructed by radii going to B and C ) is 2A, due to circle theorems. Area of each is 0.5 times the two sides times the Sine of the angle between them therefore, the area of the triangle is 0.5 r^2 (Sin 2A +Sin 2B + Sin 2C )
Once you have that the central angle is 2B, you could also extend the radius from C to a diameter and then use the fact that a triangle with one side a diameter is right. The angle next to b is 90 - B (because the central angle is 2B and the triangle is isosceles) and so the angle opposite b is also B, so since the hypotenuse is 2r you get b = 2r sin B.
You could have used S=a*b*c/4R. From sine rule we have a/sinA= b/sinB=c/sinC= 2R From there we have a=2RsinA,b=2RsinB ,c=2RsinC S=a*b*c/4R=2RsinA*2RsinB*2RsinC/4R=2R^2sinAsinBsinC
If you have the values if the three angles and r you can get the area of the three circular segments of the circle (AC, CB, BA) then take it out of the area of the whole circle in easy four steps, in another hand that's enjoyable good job 💜
At the point where you've shown the area of a triangle as half the product of any two sides, times the sine of the included angle; and you've also divided the triangle into 3 (isosceles) subtriangles, joined at the center, O, of the circle; and you have all 3 apex angles of those; you can just apply that area formula to all 3 isosceles triangles, and get Area = ½r²[sin(2A) + sin(2B) + sin(2C)] Note that this works even when ∆ABC is obtuse, so that O is outside it. All that happens then, is that the obtuse angle contributes a negative area to ∆ABC, which is correct. So the really interesting thing here is, that when angles A + B + C = π, then from your result and this one, we get the identity: sin(2A) + sin(2B) + sin(2C) = 4 sin(A) sin(B) sin(C) Fred
Use inscribed angles to find angles from the circumcenter to each pair of points. Then write A=(r, 0), B=(r cos theta1, r sin theta1), and C = (r cos theta2, r sin theta2) and then just use the shoelace determinant to get it fast.
Here's how I'd do it. The triangle equals the circle minus three slices. The area of slice a is r²(2A - sin 2A)/2. Similarly for slices b and c. Subtract all three from the area of the circle, the r²(A+B+C) cancels the area of the circle, and you're left with r²(sin 2A + sin 2B + sin 2C)/2.
you could have done this withouut doing law of cosines by considering the triangle AOB. there it is an isosceles triangle. with an angle between them is 2B. now we draw a angle bisector to the angle2B. it bisects the opposite side perpendicularly and equally. so now take sin(B) which would be equal to b/2r. so, 2r ×sin(B)=b
Well … because I am not as smart, I found another solution that works, fairly abstractly. From each point A, B and C, draw a radius to center point O. Each length is 'radius' or 𝒓 if you prefer. Because of the central-vs-edge angle theorem, we can say that № 1.1: 2∠CAB = ∠COB; № 1.2: 2∠ACB = ∠AOB; № 1.3: 2∠BAC = ∠BOC; Alright, those central angles are twice the measure of the perimeter angles A, B an C, respectively. The next step however, is central to having this work… № 2.1: ∠AOB divided by 2 is 2∠ACB/2 which is ∠C Which is the same as the others № 2.2: ∠BOC divided by 2 is 2∠BAC/2 which is ∠A № 2.3: ∠AOC divided by 2 is 2∠ABC/2 which is ∠B however, each of the corresponding segments AB, BC and AC are each divided in half with those half-angles. Since the area of each sub-triangle is ½BH, then just need to find the B and H for each: № 3.1: AB = 2 × r cos C № 3.2: BC = 2 × r cos A № 3.3: AC = 2 × r cos B And the heights of each respective are № 4.1: O-to-AB = 2 × r sin C № 3.2: O-to-BC = 2 × r sin A № 3.3: O-to-AC = 2 × r sin B Then the triangles come together Area △ AOB = ½ BH = ½ × 2 × r cos C • r sin C = r² cos C sin C Area △ BOC = ½ BH = ½ × 2 × r cos A • r sin A = r² cos A sin A Area △ AOC = ½ BH = ½ × 2 × r cos B • r sin B = r² cos B sin B Which reduce, each, to Area △ ABC = r² ( cos C sin C + cos B sin B + cos A sin A ); And that is where I would be temped to stop. However, it is not as nice a form as the one RedPenBlackPen came up with. TESTING it with a whole lot of randomly chosen angles showed that the above is also the same as Area △ ABC = 2 r² sin A sin B sin C; So, there we are. Two different unequally memorable solutions. ⋅-⋅-⋅ Just saying, ⋅-⋅-⋅ ⋅-=≡ GoatGuy ✓ ≡=-⋅
i found a triangle area formula if you know a side length and all 3 angles s = side length a1 = an angle adjacent to side s a2 = the other angle adjacent to side s o = the angle opposite side s formula : s^2 sin(a1) sin(a2) / 2 sin(o)
If r*sin A•sin B•sin C = π Then P of tringle is 2πR and the area of tringle =area of circel when R=π/(sin A• •sin B•sin C).? Its could be really so cool
Haven't watched the video yet For the proof of the equilateral triangle being the biggest, I would say that for any chord in the circle, the point which will make the triangle with the largest area has to be the furthest away from the line so thet the height is the biggest From then, you can choose a random chord and start iterating, which will result in all of the heights in the triangle being equal due to symmetry, which means the triangle is equilateral
This was a pretty long approach honestly.. It's way way simpler if you use the "chord theorem" (that is, if an angle on a circumference alpha is opposite to a chord L then L = 2r*sin(alpha) ). Hence one has a = 2r*sin(A), b=2r*sin(B) and know since the height with respect to a is h = b*sin(C) we have area = 0.5*a*h = 0.5*2*r*sin(A)*2*r*sin(B)*sin(C) = 2*(r^2)*sin(A)*sin(B)*sin(C). It's almost a 1 line proof
Hi blackpenredpen Can you help me with this problem? There is an isosceles triangle, in which the sine of the angle from the base is 3 time bigger that cosine of the angle from the top. What's the value of the sine of the angle from the base?
Sir when I solved I got expression like this Half of. R square (sin 2A +sin2B +sin2C) Is it correct I just use formula area of triangle = half xy sin theta where theta is angle between x and y
Now that you know the maximum triangle area possible in a full circle....... What about a semi-circle? Or maybe we can generalise.....if we have a sector of angle theta radians....what is the area of the largest triangle possible?
Hello!! Awesome video once again. However, we can do this much simpler!! Draw in the radii. The angles at the centre will be twice the angles at the circumference. So, total area is: A= ((r^2)/2)(sin(2A) + sin(2B) + sin(2C)) Isn't this nicer!?
Sunny Dama it's certainly faster, but the result he got looks neater. When you have formulas and you work through you want to get a neater result. And rather than use trigonometric identities to convert your answer to his, perhaps he wanted to work with cosine rule to benefit viewers
Shreyas Sarangi , actually using the cosine rule was unnecessarily complicated. Much simpler would have been to drop a perpendicular to bisect the iscoceles triangle into two right-angled triangles giving sin B = b/2 / r immediately. On the other hand, if you combine the two methods, you get a neat proof of his formula for cos 2B.
the double angle formula for cosine: cos(2b) = cos^2 (b) - sin^2 (b). then use the pythagorean identity to rewrite cos^2 (b) as 1 - sin^2 (b), so cos(2b) = 1 - 2sin^2 (b), which is the identity he used (the way you have it written isn't true, the left hand side should just be cos(2b). if you want more of these, google trig identities
From AM-GM inequality (since sin alfa, sin beta, sin gamma are positive) we have that the product of those sines is less than or equal to (sin alfa + sin beta + sin gamma)^3 / 27 and because sine is concave on the interval (0, pi) we have from Jensen's inequality that sin alfa + sin beta + sin gamma
The section 4:20 to 10:28 where you calculate b given the angle B and the radius r was needlessly complex. No need to invoke the law of the cosigns and complex algebra. Let Geometry do your work for you! Look at your diagrams! Let the center be labeled D. The triangle ADC is an isosceles triangle, have 2 sides of length r. Bisect the angle at D. This will split your isosceles triangle into 2 right triangles. The angle at the center will have measure B, because you have already noted that angle ADC has measure 2B because of central angle theorem. The hypotenuse of these right triangles is length r. The side opposite this angle has length r sin(B). Because right triangle and definition of sine. But the base is 2 of these segments. Thus the length of the base is 2r sin(B). This is much easier than fooling around with the law of the cosigns and algebra!
I forgot there's no autocorrect on whiteboard...
How to find average area of above inscribed triangle in circle of radius r?
xD
blackpenredpen
Parker Triangle
mega crossover event
Here BPRP I thought I should let you know i found two other methods to solving this. You can check here:
cdchester.co.uk/wp-content/uploads/2018/06/Area-of-an-Inscribed-Triangle-4-Methods.pdf
*English* is important but *Math* is importanter.
The englisher you speak, the gooder you become.
Because Math is the language of universe and nature !!
İf you don't know English, You don't understant people speak english only ;)
But You don't know Mats, You don't understant anything !!!
Chris Sándor Kacsó Math is important but english is 1 times more important.
@@supercool1312 OMEGALUL
*Importanterer
Also, Area= (1/2) r^2 [sin(2A)+sin(2B)+sin(2C)]
Which means If A+B+C=180, then *sin(2A)+sin(2B)+sin(2C) =4sin(A)sin(B)sin(C)*
Coool
I got that result as well by finding the area of each isosceles triangle and adding up the results
@@supertester23
RHS = 2sin(A)sin(B)(2sin(C)) (some manipulation of original RHS 4sin(A)sin(B)sin(C) where 2^2 = 4
= 2sin(C)(cos(A-B) - cos(A+B)) (product to sum, 2sin(A)sin(B) = cos(A-B) - cos(A+B))
= 2sin(C)cos((A-B)) - 2sin(C)cos((A+B)) (expansion)
= sin(C+(A-B)) + sin(C - (A-B)) - sin(C+A+B) - sin(C-A-B) (product to sum once again)
= sin(C + A - B) + sin(C - A + B) - sin(A + B + C) - sin(C - (A+B)) (expansion of input of sin())
= sin(pi - B - B) + sin(A + B + C - 2A) - sin(pi) - sin(C - (pi - C)) (using the fact A + B + C = pi)
= sin(pi - 2B) + sin(pi - 2A) - 0 - sin(C - pi + C) (sin(pi) = 0 and using the fact A + B + C = pi and expansion)
= sin(2B) + sin(2A) - sin(2C - pi) (sine is positive in the second quadrant)
= sin(2A) + sin(2B) + sin(pi - 2C) (sine is an odd function)
= sin(2A) + sin(2B) + sin(2C) (sin(pi-x) = sin(x))
= LHS
Alternatively, use the fact that BPRP's method and the method Heliocentric used to find the area must give the same area as they are the same triangle...
so (1/2)r^2[sin(2A) + Sin(2B) + sin(2C)] = 2r^2sin(A)sin(B)sin(C)
which means sin(2A)+sin(2B)+sin(2C) = 4sin(A)sin(B)sin(C)
of course, if A + B + C = 180 degrees since we are dealing with triangles.
@@supertester23 Since this formula using the double sines gives the area, we know it is equal to the area BPRP derived in the video. By cancelling r^2 and multiplying by 2 you arrange to get the equality OP derived, and we have this true only for A+B+C = 180 because this is only true when the angles are angles of a triangle, which sum to 180.
تمرين جميل جيد. رسم واضح. شرح واضح مرتب. شكرا جزيلا لكم والله يحفظكم ويرعاكم ويحميكم جميعا. تحياتنا لكم من غزة فلسطين .
You can do this alternative way to make things faster :
Step 1 : by using the law of sin.
For any triangle, the area will be 0.5*b*c*sin(A) where A is the angle between side b and c
Step 2 : by splitting the the isosceles triangle in two
the base and the angle will be split in exact half. Simply 0.5*b = r*sin(B)
Drestanto , I noticed that by combining the two methods we get a simple proof of the double angle formula for cos2A used in BPRP's method.
you'd have to prove that the triangle is indeed isosceles first then.
Wow, then, in this time we can use the law of sine, a/sin(A)=b/sin(B)=c/sin(C)=2r, now formula changes like S/abc=2r^2 sin(A)/a sin(B)/b sin(C)/c=2r^2 (1/2r)^3=1/4r, so we can know that S=abc/4r ! :)
That is really nice. This makes me miss school level geometry. Great video my friend. Keep up the good work.
Genius at work....love your channel.
Balaji Shankar thank you!
Another Way:
Use Extended Law of Sines:
a/sinA=b/sinB=c/sin C=2R
And make appropriate substitutions for b, c.
Ah!!! That's very nice!
blackpenredpen Yeah, It comes in pretty handy in Olympiads.
You can use it to prove that abc/4R formula as well.
I did the exact same thing you are going to do in the future, finding the largest inscribed triangle for any circumference, and it was a hell of an exercise! Good to see you started up just like I did a while back, and I'm very eager to see if you do anything better than I did, and I sincerely hope you destroy my proof so that I can learn from my mistakes! Keep making these awesome videos!
When you are a math god, but can't spell similarly lol
great video! amazing as always :)
He also thinks 3 is smaller than 2
AndDiracisHisProphet sometimes
maybe depends on the metric?
I would really like to be a math god than not being able to spell
LOL
Very, very satisfying. Thanks!!
Man , You are awesome. Please upload more geometric video like euclidian, Non-euclidian and other geometry.
I am ur biggest fan. The way u solve questions....it’s amazing thanks for coming on CZcams. And for this video ....thanks. This would help me in many olympiads in my 10th grade
In a circle of radius 1 , for any inscribed triangle with sides length a , b , c We have : (a^4) + (b^4) + (c^4) - 2 * ( (a^2)*(b^2) + (a^2)*(c^2) + (b^2)*(c^2) ) + (a^2)*(b^2)*(c^2) = 0
Math is love, math is life😍
Thanks a lot bro you really helped me with my math homework and also helped me widen my interest in mathematics and
also additional mathematics which is like a subject in IGCSE cambridge.
Thanks
To solve b, you could also break triangle OAC into 2 equal pieces and calculate the area of one of the pieces(little right triangle) 2 ways. You will get 1/4 bh = 1/2 rhsinB, which is b = 2rsinB, a bit easier than law of cosine.
steve* uses blue
me* unsubscribes
User unknown :(
blackpenredpen come on, you know I can't do that ))
: )
sorry for the late reply, i just saw ur reply
Maths is always so beautiful with you sir
Marvelous!
Bro really rocking the supreme
I love Lagrange Multipliers.
Not Your Average Nothing me too. Coming tmr!
Very nice presentation.
Much quicker to do like this:
It's known that abc=4R△ where a, b, and c are the side-lengths. -- (i)
It's also known that a=2RsinA, b=2RsinB, and c=2RsinC
Using these in (i), we get:
(2RsinA)(2RsinB)(2RsinC)=4R△
Or, △=(2RsinA)(2RsinB)(2RsinC)/4R=2R²sinAsinBsinC
Hello Steve, I've been subscribed to your channel for a while and I really had to thank you for everything I learned here. I'm taking calculations for the first time and their videos are very useful, I really love them. Please keep doing it and keep that good mood!
PS You are very close to reaching 100k subscribers ... I wonder what you will do to celebrate :')
Sorry 4 bad english :P
gimme dat booty thank you so much for your nice comment. I have a plan for my 100k special. :)
To celebrate bprp is going to find all 100k roots of the polynomal x^100000 = 1
What an interesting and pleasant result.
Beautiful! I don't like planimetrics very much but this is outstanding!
I think this is a pretty cool derivation for an expression for an area of a triangle inscribed in a circle. I just want to make one note about it; when you use the Law of Cosines in the triangle with vertices A, C, and the center of the circle, you assume that angle 2B is an angle in a triangle. This assumption translates to an assumption that 0 < 2B < 180 degrees, which is equivalent to 0 < B < 90 degrees. Therefore, the steps you have used in the derivation are rigorous only for angle B being acute, and since you argue the same things can be done with angle C to find an expression for c with r in it, the derivation also is only rigorous for the assumption as well that angle C is acute. I am pretty sure the final formula works for any triangle inscribed in the circle, but I think that there are a couple of additional cases to the problem to consider for doing the derivation by finding the sides of the triangle ABC you found, angle B being right or obtuse while angle C is still acute. I think the geometry and trigonometry of the derivation probably are slightly different to prove the final formula for angle B as right or obtuse. I hope you find this useful.
You're great!
Arc AC was supposed to be the 2B since B is inscribed about the circle. By the central angle theorem if O is the center then
By using law of sines,
Once you get to Area = 1/2 bcsinA
b/sinB = c/sinC = 2r
b=2r sinB c=2rsinC
Area = 1/2 4r² sinB sinC sinA
= 2r²sinAsinBsinC
i like math,physic and chemistry but i like physics more but bro your channel is superb for math ,,👍👍👍👌
Fav math teacher 😃
damn, I got so close I got
r^2 * [sin(2A) + sin(2B) + sin(2C)]
That said, wouldn't the formula i got be simpler for calculus?
Ruby Duby, you're not far away. Use the fact that A+B+C=pi with the sum to product identity for sine and sine. Combine and simplify, you get 4sinAsinBsinC. Substitute back in your original expression and that's 2r^2.sinAsinBsinC
nice :)
On a side note, i think if we equate the two formulas, we should end up getting a nice little property between angles.
2sinAsinBsinC = sin2A + sin2B + sin2C
Itd be cool if this actually worked for any polygon besides triangle.
Ruby Duby it's actually your furmula multiplied by 1/2
It's actually the same thing and it is correct.
Skip the law of cosines. To calculate b, drop a perpendicular from the center (D) to chord b. Because ADC is isocelese, CAD = ADC. Right angles at crossing, imply angle on each side of perpendicular are equal and therefore b. Then opposite side is clearly 2 sin b. Same for c.
Construct a radius to each corner. The angle 'opposite' A (angle constructed by radii going to B and C ) is 2A, due to circle theorems. Area of each is 0.5 times the two sides times the Sine of the angle between them therefore, the area of the triangle is 0.5 r^2 (Sin 2A +Sin 2B + Sin 2C )
Once you have that the central angle is 2B, you could also extend the radius from C to a diameter and then use the fact that a triangle with one side a diameter is right. The angle next to b is 90 - B (because the central angle is 2B and the triangle is isosceles) and so the angle opposite b is also B, so since the hypotenuse is 2r you get b = 2r sin B.
7:15 That's vector subtraction
MIND BLOWN
I love this video
Brilliant
You could have used S=a*b*c/4R.
From sine rule we have a/sinA= b/sinB=c/sinC= 2R
From there we have a=2RsinA,b=2RsinB ,c=2RsinC
S=a*b*c/4R=2RsinA*2RsinB*2RsinC/4R=2R^2sinAsinBsinC
If you have the values if the three angles and r you can get the area of the three circular segments of the circle (AC, CB, BA) then take it out of the area of the whole circle in easy four steps, in another hand that's enjoyable good job 💜
At the point where you've shown the area of a triangle as half the product of any two sides, times the sine of the included angle;
and you've also divided the triangle into 3 (isosceles) subtriangles, joined at the center, O, of the circle;
and you have all 3 apex angles of those;
you can just apply that area formula to all 3 isosceles triangles, and get
Area = ½r²[sin(2A) + sin(2B) + sin(2C)]
Note that this works even when ∆ABC is obtuse, so that O is outside it. All that happens then, is that the obtuse angle contributes a negative area to ∆ABC, which is correct.
So the really interesting thing here is, that when angles A + B + C = π, then from your result and this one, we get the identity:
sin(2A) + sin(2B) + sin(2C) = 4 sin(A) sin(B) sin(C)
Fred
More geometry and Calc III PLEASE!!!!
Blackpenredpen uses bluepen(*´∇｀*)
And Ur explaination is just like ur smile.
Use inscribed angles to find angles from the circumcenter to each pair of points. Then write A=(r, 0), B=(r cos theta1, r sin theta1), and C = (r cos theta2, r sin theta2) and then just use the shoelace determinant to get it fast.
soooo good!
Love your Videos! My first Intuition would have been to use polar cordinates. Maybe you'll find the time to cover that in an other video :)
This man loves his video's comments.
Math rules and you rock!
very nice!
Here's how I'd do it. The triangle equals the circle minus three slices. The area of slice a is r²(2A - sin 2A)/2. Similarly for slices b and c. Subtract all three from the area of the circle, the r²(A+B+C) cancels the area of the circle, and you're left with r²(sin 2A + sin 2B + sin 2C)/2.
you could have done this withouut doing law of cosines by considering the triangle AOB. there it is an isosceles triangle. with an angle between them is 2B. now we draw a angle bisector to the angle2B. it bisects the opposite side perpendicularly and equally. so now take sin(B) which would be equal to b/2r. so, 2r ×sin(B)=b
Furthermore, because it is equilateral triangle, angles A,B & C are same. So it can be SinA^3.
I salutes your brain . I am from india
You are the Wizard of mathematics.
Prove the central angle please I love your videos 🎊🎊💪🏼
XmatterX it's already in the description :)
If triangle is on the plane in Euclidean space we can get rid of one angle
Area = 2r^2sinAsinBsin(A+B)
Side length ‘a’ be like
''Why I'm still here?! Just to suffer''
Well … because I am not as smart, I found another solution that works, fairly abstractly.
From each point A, B and C, draw a radius to center point O. Each length is 'radius' or 𝒓 if you prefer.
Because of the central-vs-edge angle theorem, we can say that
№ 1.1: 2∠CAB = ∠COB;
№ 1.2: 2∠ACB = ∠AOB;
№ 1.3: 2∠BAC = ∠BOC;
Alright, those central angles are twice the measure of the perimeter angles A, B an C, respectively. The next step however, is central to having this work…
№ 2.1: ∠AOB divided by 2 is 2∠ACB/2 which is ∠C
Which is the same as the others
№ 2.2: ∠BOC divided by 2 is 2∠BAC/2 which is ∠A
№ 2.3: ∠AOC divided by 2 is 2∠ABC/2 which is ∠B
however, each of the corresponding segments AB, BC and AC are each divided in half with those half-angles.
Since the area of each sub-triangle is ½BH, then just need to find the B and H for each:
№ 3.1: AB = 2 × r cos C
№ 3.2: BC = 2 × r cos A
№ 3.3: AC = 2 × r cos B
And the heights of each respective are
№ 4.1: O-to-AB = 2 × r sin C
№ 3.2: O-to-BC = 2 × r sin A
№ 3.3: O-to-AC = 2 × r sin B
Then the triangles come together
Area △ AOB = ½ BH = ½ × 2 × r cos C • r sin C = r² cos C sin C
Area △ BOC = ½ BH = ½ × 2 × r cos A • r sin A = r² cos A sin A
Area △ AOC = ½ BH = ½ × 2 × r cos B • r sin B = r² cos B sin B
Which reduce, each, to
Area △ ABC = r² ( cos C sin C + cos B sin B + cos A sin A );
And that is where I would be temped to stop. However, it is not as nice a form as the one RedPenBlackPen came up with. TESTING it with a whole lot of randomly chosen angles showed that the above is also the same as
Area △ ABC = 2 r² sin A sin B sin C;
So, there we are.
Two different unequally memorable solutions.
⋅-⋅-⋅ Just saying, ⋅-⋅-⋅
⋅-=≡ GoatGuy ✓ ≡=-⋅
The same result can be obtained by simply using Area=(1/2)*a*b*sinC and the extended sine law
When a video math is great, it's no problem the language!
Nice video
Little idea to make the formula need less input: just replace the angle C by 180° - A - B, like
Area = 2r² * sin A * sin B * sin (180° - A - B)
Nice, this is the sinuses theorem, isn't it?, I remember when a was young, thanks mate. Now the Heron formula, ;-)
i found a triangle area formula if you know a side length and all 3 angles
s = side length
a1 = an angle adjacent to side s
a2 = the other angle adjacent to side s
o = the angle opposite side s
formula : s^2 sin(a1) sin(a2) / 2 sin(o)
Came be done using lami theorem and area=ABC/4r
Thanks
If r*sin A•sin B•sin C = π
Then P of tringle is 2πR and the area of tringle =area of circel when R=π/(sin A• •sin B•sin C).? Its could be really so cool
Haven't watched the video yet
For the proof of the equilateral triangle being the biggest, I would say that for any chord in the circle, the point which will make the triangle with the largest area has to be the furthest away from the line so thet the height is the biggest
From then, you can choose a random chord and start iterating, which will result in all of the heights in the triangle being equal due to symmetry, which means the triangle is equilateral
👌 very nice
Thanks sir
This was a pretty long approach honestly.. It's way way simpler if you use the "chord theorem" (that is, if an angle on a circumference alpha is opposite to a chord L then L = 2r*sin(alpha) ).
Hence one has a = 2r*sin(A), b=2r*sin(B) and know since the height with respect to a is h = b*sin(C) we have area = 0.5*a*h = 0.5*2*r*sin(A)*2*r*sin(B)*sin(C) = 2*(r^2)*sin(A)*sin(B)*sin(C).
It's almost a 1 line proof
Nicola Rares Franco good
thanks and by the way, I really like the style of your videos!
Hi blackpenredpen
Can you help me with this problem?
There is an isosceles triangle, in which the sine of the angle from the base is 3 time bigger that cosine of the angle from the top.
What's the value of the sine of the angle from the base?
Sir when I solved I got expression like this
Half of. R square (sin 2A +sin2B +sin2C)
Is it correct I just use formula area of triangle = half xy sin theta where theta is angle between x and y
how can we find area of the inscribed triangle in the trapezoid??
given vertices are: (8,6) (4,0) and (0,3)
Now that you know the maximum triangle area possible in a full circle.......
What about a semi-circle? Or maybe we can generalise.....if we have a sector of angle theta radians....what is the area of the largest triangle possible?
Nice video! Similarly :D
So, find circumcenter first, and you don't need the circle. Cool. :)
msolec2000 Point of concurrence of sides' perpendicular bisectors? Something like that?
Well with only r you can't tell what's the area. You must know any 3 of (the side lengths or the angles).
You could simplify the formula, using Sin law.
Hello!!
Awesome video once again.
However, we can do this much simpler!!
Draw in the radii. The angles at the centre will be twice the angles at the circumference. So, total area is:
A= ((r^2)/2)(sin(2A) + sin(2B) + sin(2C))
Isn't this nicer!?
Sunny Dama it's certainly faster, but the result he got looks neater. When you have formulas and you work through you want to get a neater result. And rather than use trigonometric identities to convert your answer to his, perhaps he wanted to work with cosine rule to benefit viewers
Shreyas Sarangi , actually using the cosine rule was unnecessarily complicated. Much simpler would have been to drop a perpendicular to bisect the iscoceles triangle into two right-angled triangles giving sin B = b/2 / r immediately. On the other hand, if you combine the two methods, you get a neat proof of his formula for cos 2B.
Did you edit the video because there is an excessive amount of motionblurr when you turn your head.
law of cosine was overkill a bit :p you can use twice the projection with sin(B) on b. :)
Prove Morley's theorem.
Thank You
Great video. Can someone please explain to me why 1-cos(2b)=1-2sin^2(B)? Where can I find more manipulations like that?
the double angle formula for cosine: cos(2b) = cos^2 (b) - sin^2 (b). then use the pythagorean identity to rewrite cos^2 (b) as 1 - sin^2 (b), so cos(2b) = 1 - 2sin^2 (b), which is the identity he used (the way you have it written isn't true, the left hand side should just be cos(2b). if you want more of these, google trig identities
Area is also equal to:
A=abc/4r
"Always Be Careful 4 Rapist"
Hey nice free hand circle
what's the largest area of a triangle inscribed in a circle with radius r?
That's coming up tomorrow.
It has to be equilateral tho.
A = ¾ r² √3
From AM-GM inequality (since sin alfa, sin beta, sin gamma are positive) we have that the product of those sines is less than or equal to (sin alfa + sin beta + sin gamma)^3 / 27 and because sine is concave on the interval (0, pi) we have from Jensen's inequality that sin alfa + sin beta + sin gamma
Netherman 41 Nice proof :)
Can you send the link?
oh boy, this reminds me of the putnam exam question...
First!
Like your videos! :)
The section 4:20 to 10:28 where you calculate b given the angle B and the radius r was needlessly complex. No need to invoke the law of the cosigns and complex algebra. Let Geometry do your work for you! Look at your diagrams!
Let the center be labeled D. The triangle ADC is an isosceles triangle, have 2 sides of length r. Bisect the angle at D. This will split your isosceles triangle into 2 right triangles. The angle at the center will have measure B, because you have already noted that angle ADC has measure 2B because of central angle theorem. The hypotenuse of these right triangles is length r. The side opposite this angle has length r sin(B). Because right triangle and definition of sine. But the base is 2 of these segments. Thus the length of the base is 2r sin(B).
This is much easier than fooling around with the law of the cosigns and algebra!
actually do u have a curriculum of what u teach in ur vids?
The most confusing part is... why did he use the law of cosines to prove the law of sines? He could have used it right away, couldn't he?
No NO *NO!* you can't leave me hanging like that... D: Post that next damn video already!
edit: found it, thanks! :D
LOLLLLLL
a/sinA= b/sinB= c/sinC= 2R
short form for similarly used to be "///ly" don't know if this is widely used though? saves you worrying about spelling !!!!
And C is also Pi-A-B.
I just thought you hold a bomb. Gosh
You have assumed that the interior of the triangle contains the centre of the circle. Proof may have to be adapted if this is not the case.
In the central angle theorem video he has proven it's also true when the center is outside the inscribed angle.