MathDB

We are given an acute-angled triangle

ABC

and a random point

X

in its interior, different from the centre of the circumcircle

k

of the triangle. The lines

AX,BX

and

CX

intersect

k

for a second time in the points

A_1,B_1

and

C_1

respectively. Let

A_2,B_2

and

C_2

be the points that are symmetric of

A_1,B_1

and

C_1

in respect to

BC,AC

and

AB

respectively. Prove that the circumcircle of the triangle

A_2,B_2

and

C_2

passes through a constant point that does not depend on the choice of

X

Problem Statement