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Double parallels

Source: Mexico National Olympiad 2011 Problem 6

June 22, 2014

geometry proposedgeometry

Problem Statement

Let

\mathcal{C}_1

and

\mathcal{C}_2

be two circumferences intersecting at points

A

and

B

. Let

C

be a point on line

AB

such that

B

lies between

A

and

C

. Let

P

and

Q

be points on

\mathcal{C}_1

and

\mathcal{C}_2

respectively such that

CP

and

CQ

are tangent to

\mathcal{C}_1

and

\mathcal{C}_2

respectively,

P

is not inside

\mathcal{C}_2

and

Q

is not inside

\mathcal{C}_1

. Line

PQ

cuts

\mathcal{C}_1

at

R

and

\mathcal{C}_2

at

S

, both points different from

P

,

Q

and

B

. Suppose

CR

cuts

\mathcal{C}_1

again at

X

and

CS

cuts

\mathcal{C}_2

again at

Y

. Let

Z

be a point on line

XY

. Prove

SZ

is parallel to

QX

if and only if

PZ

is parallel to

RX

.

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