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Determine cosine of angle

Source: Canadian Mathematical Olympiad - 1994 - Problem 4.

May 13, 2011

Problem Statement

Let

AB

be a diameter of a circle

\Omega

and

P

be any point not on the line through

AB

. Suppose that the line through

PA

cuts

\Omega

again at

U

, and the line through

PB

cuts

\Omega

at

V

. Note that in case of tangency,

U

may coincide with

A

or

V

might coincide with

B

. Also, if

P

is on

\Omega

then

P=U=V

. Suppose that

|PU|=s|PA|

and

|PV|=t|PB|

for some

0\le s,t\in \mathbb{R}

. Determine

\cos \angle APB

in terms of

s,t

.

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