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 2 of 5: Use power at M

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MD^2=MC^2=MG\cdot MP

Detailed analysis

Since $M$ lies on the tangent to $\Gamma$ at $C$ , the power of $M$ with respect to $\Gamma$ gives $MC^2=MG\cdot MP$ . With $MC=MD$ , we get $MD^2=MG\cdot MP$ , so $MD$ is tangent to the circumcircle of triangle $DGP$ .