2001 USAMO Problems/Problem 2
From AoPSWiki
Problem
Let 
be a triangle and let 
be its incircle. Denote by 
and 
the points where is tangent to sides 
and 
, respectively. Denote by 
and 
the points on sides and , respectively, such that 
and 
, and denote by 
the point of intersection of segments 
and 
. Circle intersects segment at two points, the closer of which to the vertex 
is denoted by 
. Prove that 
.
Solution
It is well known that the excircle opposite is tangent to 
at the point . (Proof: let the points of tangency of the excircle with the lines 
be 
respectively. Then 
. It follows that 
, and 
, so 
.)
Now consider the homothety that carries the incircle of 
to its excircle. The homothety also carries to (since 
are collinear), and carries the tangency points to 
. It follows that 
.
By Menelaus' Theorem on 
with segment 
, it follows that 
. It easily follows that . 
See also
| 2001 USAMO (Problems • Resources: AoPS | ML) | ||
| Preceded by Problem 1 | 1 • 2 • 3 • 4 • 5 • 6 | Followed by Problem 3 |
