Assignment 8.

13. The internal angle bisectors of triangle ABC are extended to meet the circumcircle at points L, M, and N, respectively. Find the angles of triangle LMN in terms of the angles A, B, and C. Does your result hold only for acute triangles?

First, explore this problem using GSP file after downloading. To download the file, Click HERE!

Drag the vertices  and observe the angles of given triangle.

My conjecture is

.

Now, I will prove my conjecture.

Since (because the ray AL is the bisector of angle BAC),

the length of arc BL = the length of arc CL.

By the property that angles inscribed in a circle are the same,

.

Similarly,

Thus,

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