MathDB

An acute-angled triangle

ABC

is given, through the vertices

B

and

C

of which a circle

\Omega

A \notin \Omega

, is drawn. We consider all points

P \in \Omega

, that do not lie on none of the lines

AB

and

AC

and for which the common tangents of the circumscribed circles of triangles

APB

and

APC

are not parallel. Let

X_P

be the point of intersection of such two common tangents. a) Prove that the locus of points

X_P

lies to some two lines. b) Prove that if the circle

\Omega

passes through the orthocenter of the triangle

ABC

, then one of these lines is the line

BC

Problem Statement