MathDB

The incircle of a triangle

ABC

centered at

I

touches the sides

BC, CA

, and

AB

at points

A_1, B_1,

and

C_1

respectively. The excircle centered at

J

touches the side

AC

at point

B_2

and touches the extensions of

AB, BC

at points

C_2, A_2

respectively. Let the lines

IB_2

and

JB_1

meet at point

X

, the lines

IC_2

and

JC_1

meet at point

Y

, the lines

IA_2

and

JA_1

meet at point

Z

. Prove that if one of points

X, Y, Z

lies on the incircle then two remaining points also lie on it.

Problem Statement