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 3 of 5: Obtain the angle relation

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\angle DGP=\angle EDP

Detailed analysis

The tangent-chord theorem for the circumcircle of DGP gives angle DGP=angle EDP, because E,D,M are collinear. If P is on the same side of BC as G, combining this with angle ABP yields angle EDP+angle ABP=180 degrees; if it is on the other side, the directed-angle equalities give the same cyclicity conclusion.