MathDB

Let

ABC

be an acute triangle and suppose points

A, A_b, B_a, B, B_c, C_b, C, C_a,

and

A_c

lie on its perimeter in this order. Let

A_1 \neq A

be the second intersection point of the circumcircles of triangles

AA_bC_a

and

AA_cB_a

. Analogously,

B_1 \neq B

is the second intersection point of the circumcircles of triangles

BB_cA_b

and

BB_aC_b

, and

C_1 \neq C

is the second intersection point of the circumcircles of triangles

CC_aB_c

and

CC_bA_c

. Suppose that the points

A_1, B_1,

and

C_1

are all distinct, lie inside the triangle

ABC

, and do not lie on a single line. Prove that lines

AA_1, BB_1, CC_1,

and the circumcircle of triangle

A_1B_1C_1

all pass through a common point.
Josef Tkadlec (Czech Republic), Patrik Bak (Slovakia)

Problem Statement