Unlock the Secrets of Incenter and Excenter with this Geometry Lemma

Let ABC be a triangle with incenter I, line AI meets the circumcircle of △ABC again at L. We let I_A be the reflection of I over L. Then, we have two facts, first fact is (a) the points I, B, C and I_A lie on a circle with diameter II_A and center L. In particular, LI = LB = LC = LI_A. The second fact is (b) lines BI_A and CI_A bisect the exterior angles of △ABC. In other words, they are the exterior angle bisectors of ∠B and ∠C.
Can you prove this lemma? Well, this is a useful lemma in Euclidean geometry, especially if you are solving Math Olympiad geometry problems and I want to point out a great book in learning it which is the ”Euclidean Geometry in Mathematical Olympiad” by Evan Chen, there is a link below to this book.
🔥Subscribe to 1Psi3Colour:
czcams.com/users/1Psi3Colour...
Video Chapters:
0:00 The Lemma
1:09 First fact
3:31 Second fact
4:51 Outro + Subscribe!
”Euclidean Geometry in Mathematical Olympiad” by Evan Chen:
web.evanchen.cc/geombook.html
💥Check out these videos:
The integral everyone should know: • The MOST Influential I...
A deceivingly easy Putnam problem!: • Is This THE EASIEST Pr...
👉Check out these playlists:
Math Olympiad: • Math Olympiad
Math puzzles: • Math puzzles
College Math: • College Math
This video is created using Manim:
www.manim.community
👉Suggest a problem: forms.gle/hTibrUKz7QNyqoC48

[@1psi3colourmath]

This is actually one of the first non-trivial lemma I learned in Euclidean Geometry, if you didn't see this lemma before, hope that you find it interesting. I have done some geometry videos before, if you want to watch, here's one: czcams.com/video/L0LzDZYNDRw/video.html
If you find that my videos are interesting, then consider subscribing: czcams.com/users/1Psi3Colour?sub_confrimation=1

[@NaN_000]

I like geometry problems
By the way where do you get this math problem?
Thanks 🙂

[@1psi3colourmath]

The problem is from the book I mentioned in the video

[@NaN_000]

@@1psi3colourmath okey 🙂