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non-trivial first problem :) 4 concyclic points

Source: Balkan Mathematical Olympiad 2008 Problem 1

May 6, 2008

geometrycircumcircleparallelogramgeometric transformationreflectionanalytic geometrygraphing lines

Problem Statement

Given a scalene acute triangle

ABC

with

AC>BC

let

F

be the foot of the altitude from

C

. Let

P

be a point on

AB

, different from

A

so that AF\equal{}PF. Let

H,O,M

be the orthocenter, circumcenter and midpoint of

[AC]

. Let

X

be the intersection point of

BC

and

HP

. Let

Y

be the intersection point of

OM

and

FX

and let

OF

intersect

AC

at

Z

. Prove that

F,M,Y,Z

are concyclic.

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