1988 USAMO Problems/Problem 4
Problem
is a triangle with incenter
. Show that the circumcenters of
,
, and
lie on a circle whose center is the circumcenter of
.
Solution
Let the circumcenters of ,
, and
be
,
, and
, respectively. It then suffices to show that
,
,
,
,
, and
are concyclic.
We shall prove that quadrilateral is cyclic first. Let
,
, and
. Then
and
. Therefore minor arc $\arc{BIC}$ (Error compiling LaTeX. Unknown error_msg) in the circumcircle of
has a degree measure of
. This shows that
, implying that
. Therefore quadrilateral
is cyclic.
This shows that point is on the circumcircle of
. Analagous proofs show that
and
are also on the circumcircle of
, which completes the proof.
See Also
1988 USAMO (Problems • Resources) | ||
Preceded by Problem 3 |
Followed by Problem 5 | |
1 • 2 • 3 • 4 • 5 | ||
All USAMO Problems and Solutions |