2019 USAJMO Problems/Problem 3
Problem
Let
be a cyclic quadrilateral satisfying
. The diagonals of
intersect at
. Let
be a point on side
satisfying
. Show that line
bisects
.
Solution
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:
Claim:
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.
~sriraamster
The problems on this page are copyrighted by the Mathematical Association of America's American Mathematics Competitions.
See also
2019 USAJMO (Problems • Resources) | ||
Preceded by Problem 2 |
Followed by Problem 4 | |
1 • 2 • 3 • 4 • 5 • 6 | ||
All USAJMO Problems and Solutions |