Problem 3

Let

\Gamma

be the circumcircle of a triangle

ABC

. A circle passing through points

A

and

C

meets the sides

BC

and

BA

D

and

E

, respectively. The lines

AD

and

CE

meet

\Gamma

again at

G

and

H

, respectively. The tangent lines of

\Gamma

A

and

C

meet the line

DE

L

and

M

, respectively. Prove that the lines

LH

and

MG

meet at a point on

\Gamma

Step 1 of 5: Choose the point on Gamma

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P=MG\cap\Gamma,\quad P\ne G

Detailed analysis

Let P be the second point where line MG meets Gamma. The tangent-angle relations in the given configuration give angle MCD=angle CAE and angle MDC=angle CAE, hence MC=MD.