2020 AIME II Problems/Problem 15
Contents
[hide]Problem
Let be an acute scalene triangle with circumcircle
. The tangents to
at
and
intersect at
. Let
and
be the projections of
onto lines
and
, respectively. Suppose
,
, and
. Find
.
Solution
Assume to be the center of triangle
,
cross
at
, link
,
. Let
be the middle point of
and
be the middle point of
, so we have
. Since
, we have
. Notice that \angle
, so
, and this gives us
. Since
is perpendicular to
,
and
cocycle (respectively), so
and
. So
, so
, which yields
So same we have
. Apply Ptolemy theorem in
we have
, and use Pythagoras theorem we have
. Same in
and triangle
we have
and
. Solve this for
and
and submit into the equation about
, we can obtain the result
.
-Fanyuchen20020715
Video Solution
https://youtu.be/bz5N-jI2e0U?t=710
See Also
2020 AIME II (Problems • Answer Key • Resources) | ||
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