MathDB

Concurrent Lines Problem from a TST

Source: 2018 China TST 3 Day 2 Problem 5

March 27, 2018

concurrentgeometrycircumcircle

Problem Statement

Let

ABC

be a triangle with

\angle BAC > 90 ^{\circ}

, and let

O

be its circumcenter and

\omega

be its circumcircle. The tangent line of

\omega

at

A

intersects the tangent line of

\omega

at

B

and

C

respectively at point

P

and

Q

. Let

D,E

be the feet of the altitudes from

P,Q

onto

BC

, respectively.

F,G

are two points on

\overline{PQ}

different from

A

, so that

A,F,B,E

and

A,G,C,D

are both concyclic. Let M be the midpoint of

\overline{DE}

. Prove that

DF,OM,EG

are concurrent.

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