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Equality of perimeters

Source: Central American Olympiad 2003, Problem 4

June 1, 2007

geometryrectangletrapezoid

Problem Statement

S_{1}

and

S_{2}

are two circles that intersect at distinct points

P

and

Q

.

\ell_{1}

and

\ell_{2}

are two parallel lines through

P

and

Q

.

\ell_{1}

intersects

S_{1}

and

S_{2}

at points

A_{1}

and

A_{2}

, different from

P

, respectively.

\ell_{2}

intersects

S_{1}

and

S_{2}

at points

B_{1}

and

B_{2}

, different from

Q

, respectively. Show that the perimeters of the triangles

A_{1}QA_{2}

and

B_{1}PB_{2}

are equal.

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