2015 USAJMO Problems/Problem 5
Problem
Let be a cyclic quadrilateral. Prove that there exists a point
on segment
such that
and
if and only if there exists a point
on segment
such that
and
.
Solution
Note that lines are isogonal in
, so an inversion centered at
with power
composed with a reflection about the angle bisector of
swaps the pairs
and
. Thus,
so that
is a harmonic quadrilateral. By symmetry, if
exists, then
. We have shown the two conditions are equivalent, whence both directions follow