MathDB

Let

ABC

be a triangle, and let

C^{\prime}

and

A^{\prime}

be the feet of its altitudes issuing from the vertices

C

and

A

, respectively. Denote by

P

the midpoint of the segment

C^{\prime}A^{\prime}

. The circumcircles of triangles

AC^{\prime}P

and

CA^{\prime}P

have a common point apart from

P

; denote this common point by

Q

. Prove that: (a) The point

Q

lies on the circumcircle of the triangle

ABC

. (b) The line

PQ

passes through the point

B

. (c) We have

\frac{AQ}{CQ}=\frac{AB}{CB}

. Darij

Problem Statement