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two collinearities wanted

Source: Norwegian Mathematical Olympiad 1998 - Abel Competition p4

February 11, 2020

collineargeometry

Problem Statement

Let

A,B,P

be points on a line

m

, with

P

outside the segment

\ell

. Lines

a

and

b

pass through

A

and

B

and are perpendicular to

\ell

. A line

m

through

a

, which is neither parallel nor perpendicular to

AR

, intersects

a

and

b

at

Q

and

R

, respectively. The perpendicular from

B

to

AR

meets

a

and

AR

at

P,S,T

and

U

, and the perpendicular from

A

to

BQ

meets

b

and

BQ

at

T

and

V

, respectively. (a) Prove that

P,S,T

are collinear. (b) Prove that

P,U,V

are collinear.

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