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2018 USAMO Problems/Problem 5

Revision as of 10:52, 21 April 2018 by Sujaykazi (talk | contribs) (Created page with "==Problem 5== In convex cyclic quadrilateral <math>ABCD,</math> we know that lines <math>AC</math> and <math>BD</math> intersect at <math>E,</math> lines <math>AB</math> and <...")
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Problem 5

In convex cyclic quadrilateral $ABCD,$ we know that lines $AC$ and $BD$ intersect at $E,$ lines $AB$ and $CD$ intersect at $F,$ and lines $BC$ and $DA$ intersect at $G.$ Suppose that the circumcircle of $\triangle ABE$ intersects line $CB$ at $B$ and $P$, and the circumcircle of $\triangle ADE$ intersects line $CD$ at $D$ and $Q$, where $C,B,P,G$ and $C,Q,D,F$ are collinear in that order. Prove that if lines $FP$ and $GQ$ intersect at $M$, then $\angle MAC = 90^{\circ}.$


Solution

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