MathDB

collinear wanted, 2 circles and orthocenter related

Source: OLCOMA Costa Rica National Olympiad, Final Round, 2016 Shortlist G1 day2

September 25, 2021

geometrycollinear

Problem Statement

Let

\vartriangle ABC

be acute with orthocenter

H

. Let

X

be a point on

BC

such that

B-X-C

. Let

\Gamma

be the circumscribed circle of

\vartriangle BHX

and

\Gamma_2

be the circumscribed circle of

\vartriangle CHX

. Let

E

be the intersection of

AB

with

\Gamma

, and

D

be the intersection of

AC

with

\Gamma_2

. Let

L

be the intersection of line

HD

with

\Gamma

and

J

be the intersection of line

EH

with

\Gamma_2

. Prove that points

L

,

X

, and

J

are collinear.
Notation:

A-B-C

means than points

A,B,C

are collinear in that order i.e.

B

lies between

A

and

C

.

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