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 4 of 5: Place P on the second circumcircle

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P\in( BDE )

Detailed analysis

The preceding angle relation says that quadrilateral PBDE is cyclic, so P lies on the circumcircle Gamma' of triangle BDE. The exceptional tangency case P=B is included by interpreting Gamma' as internally tangent to Gamma at B.