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 5 of 5: Repeat for LH

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LH\cap\Gamma=MG\cap\Gamma=P

Detailed analysis

Let Q be the second intersection of LH with Gamma. Repeating the same tangent, power, and cyclicity argument shows that Q also lies on Gamma'. The circles Gamma and Gamma' have the same relevant intersection point (or the same tangency point B), so Q=P. Therefore LH and MG meet at P on Gamma.