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fixed point and midpoint wanted

Source: Slovenia TST 1998 p5

February 15, 2020

midpointFixed pointgeometry

Problem Statement

On a line

p

which does not meet a circle

K

with center

O

, point

P

is taken such that

OP \perp p

. Let

X \ne P

be an arbitrary point on

p

. The tangents from

X

to

Y

touch it at

AB

and

B

. Denote by

C

and

D

the orthogonal projections of

P

on

AX

and

BX

respectively. (a) Prove that the intersection point

Y

of

AB

and

OP

is independent of the location of

X

. (b) Lines

CD

and

OP

meet at

Z

. Prove that

Z

is the midpoint of

P

.

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