Problem 3

Let

ABCD

be a cyclic convex quadrilateral and

\Gamma

be its circumcircle. Let

E

be the intersection of the diagonals

AC

and

BD

, let

L

be the center of the circle tangent to sides

AB

BC

, and

CD

, and let

M

be the midpoint of the arc

BC

\Gamma

not containing

A

and

D

. Prove that the excenter of triangle

BCE

opposite

E

lies on the line

LM

Step 4 of 8: M is the isosceles apex over BC

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\angle MBC=\angle MCB=\tfrac b2,\quad MB=MC

Detailed analysis

Since $M$ is the midpoint of arc $BC$ (measure $2b$ ) not containing $A,D$ , arcs $BM$ and $MC$ each measure $b$ , so $\angle MBC=\angle MCB=\tfrac b2$ and $MB=MC$ .