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Berkeley Math Circle: Monthly Contest 1

Geometry Difficulty 5.2 AIME, harder Prove it United States

Problem:

Let $ABCD$ be a convex quadrilateral. Assume that the incircle of triangle $ABD$ is tangent to $\overline{AB}$ , $\overline{AD}$ , $\overline{BD}$ at points $W$ , $Z$ , $K$ . Also assume that the incircle of triangle $CBD$ is tangent to $\overline{CB}$ , $\overline{CD}$ , $\overline{BD}$ at points $X$ , $Y$ , $K$ . Prove that quadrilateral $WXYZ$ is cyclic.

Solution

Solution:

From the concurrence of the Gergonne point, it follows that lines $WZ$ , $XY$ , and $BD$ concur at the harmonic conjugate $T$ of $K$ with respect to $\overline{BC}$ . (One can also see the concurrence directly by applying Ceva and Menelaus.) Then $TK^2 = TW \cdot TZ = TX \cdot TY$ , as desired.

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