2001 IMO Shortlist Problems/G8
Problem
Let be a triangle with
. Let
bisect
and let
bisect
, with
on
and
on
. If
, what are the angles of the triangle?
Solution
We will have
Let R lie on AC s.t. RAB is equilateral. Let T lie on AB extended past B such that BT = BP, and let U lie on AC such that UAT is equilateral. Since AU = AT = AB + BP = AQ + QB, we have QB = QU. As a result, we calculate . Meanwhile,
, so we have
.
Then suppose the bisector of intersects BR at X. Then since
, we have similar triangles, and by equal ratios
. Equivalently,
, so triangles BXP and BPR are similar; in particular, BX = XP.
Since TX is the bisector of , we have then that T, B, X, and P are concyclic. Then
. Solving for x, x = 40. Then
.