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Line Passing through Feet of Altitudes Intersects Circumcircle

Source: 2019 AIME II #15

March 22, 2019

AIMEAIME IIgeometrycircumcircle2019 AIME IIaltitudepower of a point

Problem Statement

In acute triangle

ABC

points

P

and

Q

are the feet of the perpendiculars from

C

to

\overline{AB}

and from

B

to

\overline{AC}

, respectively. Line

PQ

intersects the circumcircle of

\triangle ABC

in two distinct points,

X

and

m\sqrt n

. Suppose

XP=10

,

PQ=25

, and

QY=15

. The value of

AB\cdot AC

can be written in the form

m\sqrt n

where

m

and

n

are positive integers, and

n

is not divisible by the square of any prime. Find

m+n

.

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