MathDB

2021 Geo #10: radius of circumcircle

Source:

May 30, 2021

geometrycircumcircle

Problem Statement

Acute triangle

ABC

has circumcircle

\Gamma

. Let

M

be the midpoint of

BC.

Points

P

and

Q

lie on

\Gamma

so that

\angle APM = 90^{\circ}

and

Q \neq A

lies on line

AM.

Segments

PQ

and

BC

intersect at

S

. Suppose that

BS = 1, CS = 3, PQ = 8\sqrt{\tfrac{7}{37}},

and the radius of

\Gamma

is

r

. If the sum of all possible values of

r^2

can be expressed as

\tfrac ab

for relatively prime positive integers

a

and

b,

compute

100a + b

.

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