Competition problem — Geometry — Maths Olympiad Prep

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Geometry Difficulty 4.7 AIME Find the answer

Let triangle $A B C$ have $A B=5, B C=6$ , and $A C=7$ , with circumcenter $O$ . Extend ray $A B$ to point $D$ such that $B D=5$ , and extend ray $B C$ to point $E$ such that $O D=O E$ . Find $C E$ .

Your answer

A number or a short expression. Spacing and $ signs are ignored.

Solution

Because $O D=O E, D$ and $E$ have equal power with respect to the circle, so $(E C)(E B)=(D B)(D A)=50$ . Letting $E C=x$ , we have $x(x+6)=50$ , and taking the positive root gives $x=\sqrt{59}-3$ .

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