Difference between revisions of "2018 USAMO Problems/Problem 5"
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Revision as of 20:30, 6 March 2023
Problem 5
In convex cyclic quadrilateral we know that lines
and
intersect at
lines
and
intersect at
and lines
and
intersect at
Suppose that the circumcircle of
intersects line
at
and
, and the circumcircle of
intersects line
at
and
, where
and
are collinear in that order. Prove that if lines
and
intersect at
, then
Solution
so
are collinear. Furthermore, note that
is cyclic because:
Notice that since
is the intersection of
and
, it is the Miquel point of
.
Now define as the intersection of
and
. From Pappus's theorem on
that
are collinear. It’s a well known property of Miquel points that
, so it follows that
, as desired.
~AopsUser101