2022 AIME II Problems/Problem 15
Problem
Two externally tangent circles and
have centers
and
, respectively. A third circle
passing through
and
intersects
at
and
and
at
and
, as shown. Suppose that
,
,
, and
is a convex hexagon. Find the area of this hexagon.
Solution
First observe that and
. Let points
and
be the reflections of
and
, respectively, about the perpendicular bisector of
. Then quadrilaterals
and
are congruent, so hexagons
and
have the same area. Furthermore, triangles
and
are congruent, so
and quadrilateral
is an isosceles trapezoid.
Next, remark that
, so quadrilateral
is also an isosceles trapezoid; in turn,
, and similarly
. Thus, Ptolmey's theorem on
yields
, whence
. Let
. The Law of Cosines on triangle
yields
and hence
. Thus the distance between bases
and
is
(in fact,
is a
triangle with a
triangle removed), which implies the area of
is
.
Now let and
; the tangency of circles
and
implies
. Furthermore, angles
and
are opposite angles in cyclic quadrilateral
, which implies the measure of angle
is
. Therefore, the Law of Cosines applied to triangle
yields
Thus , and so the area of triangle
is
.
Thus, the area of hexagon is
.
~djmathman
See Also
2022 AIME II (Problems • Answer Key • Resources) | ||
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