2019 USAMO Problems/Problem 2
Let be a cyclic quadrilateral satisfying . The diagonals of intersect at . Let be a point on side satisfying . Show that line bisects .
Let . Also, let be the midpoint of . Note that only one point satisfies the given angle condition. With this in mind, construct with the following properties:
Proof: The conditions imply the similarities and whence as desired.
Claim: is a symmedian in
Proof: We have as desired.
Since is the isogonal conjugate of , . However implies that is the midpoint of from similar triangles, so we are done.
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