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2003 USAMO Problems/Problem 4

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Problem

Let $ABC$ be a triangle. A circle passing through $A$ and $B$ intersects segments $AC$ and $BC$ at $D$ and $E$ , respectively. Lines $AB$ and $DE$ intersect at $F$ , while lines $BD$ and $CF$ intersect at $M$ . Prove that $MF = MC$ if and only if $MB\cdot MD = MC^2$ .

Solution

by April

Take $G\in BD: \,FG\parallel CD$ . We have:

$MF = MC\Longleftrightarrow \textrm{the quadrilateral}\; CDFG\; \textrm{is a parallelogram} \\\Longleftrightarrow FD\parallel ...$

Resources

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Category: Olympiad Geometry Problems

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