Mock Geometry AIME 2011 Problems/Problem 5
Problem
In triangle
The bisector of angle
meet
at
and the circumcircle at
different from
. Calculate the value of
Solution
because they are both subscribed by arc
.
because they are both subscribed by arc
. Hence
, because
. Then
is isosceles.
Let be the foot of the perpendicular from
to
. As
is isosceles, it follows that
is the midpoint of
, and so
. From the angle bisector theorem,
. We have
. Solving this system of equations yields
. Thus,
.
because they are vertical angles. It was shown
, and so
by
similarity. Then
and so
.
Then by the Pythagorean Theorem on ,
. Also from
,
. Subtracting these equations yields
, and so
.