Difference between revisions of "2023 SSMO Team Round Problems/Problem 3"
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Latest revision as of 02:10, 3 January 2024
Problem
Let be a triangle such that
and
Let
be the circumcircle of
. Let
be on the circle such that
Let
be the point diametrically opposite of
. Let
be the point diametrically opposite
. Find the area of the quadrilateral
in terms of a mixed number
. Find
.
Solution
Note that is right with the right angle at
. This means that
is the diameter of the circle. We can divide quadrilateral
into
and
, both of which are right triangles. Mark the intersection point between BD and AC as G. We can use the fact that
and
are similar to find that
, so
by symmetry. Then,
by the Pythagorean Theorem, so the area of
is
. Since
by symmetry,
, so the area of
is
. This means that the area of the entire quadrilateral equals
, so the answer is
.
~alexanderruan