2019 USAJMO Problems/Problem 3
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:
The conditions imply the similarities and whence as desired.
Claim: is a symmedian in
Since is the isogonal conjugate of , . However implies that is the midpoint of from similar triangles, so we are done.
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