MathDB

Circumcircle passes through orthocentre [Fuhrmann extended]

Source: China TST 2006

June 18, 2006

geometrycircumcirclegeometric transformationreflectionparallelogramratioanalytic geometry

Problem Statement

Let

\omega

be the circumcircle of

\triangle{ABC}

P

is an interior point of

\triangle{ABC}

A_{1}, B_{1}, C_{1}

are the intersections of

AP, BP, CP

respectively and

A_{2}, B_{2}, C_{2}

are the symmetrical points of

A_{1}, B_{1}, C_{1}

with respect to the midpoints of side

BC, CA, AB

. Show that the circumcircle of

\triangle{A_{2}B_{2}C_{2}}

passes through the orthocentre of

\triangle{ABC}