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National and Regional Contests
Moldova Contests
EGMO TST - Moldova
2019 Moldova EGMO TST
6
6
Part of
2019 Moldova EGMO TST
Problems
(1)
Prove that the lines $PQ$ and $AB$ are perpendicular.
Source: Moldova EGMO TST 2019
7/10/2023
There is a point
T
T
T
on a circle with the radius
R
R
R
. Points
A
A{}
A
and
B
B
B
are on the tangent to the circle that goes through
T
T
T
such that they are on the same side of
T
T
T
and
T
A
ā
T
B
=
4
R
2
TA\cdot TB=4R^2
T
A
ā
TB
=
4
R
2
. The point
S
S
S
is diametrically opposed to
T
T
T
. Lines
A
S
AS
A
S
and
B
S
BS
BS
intersect the circle again in
P
P{}
P
and
Q
Q{}
Q
. Prove that the lines
P
Q
PQ
PQ
and
A
B
AB{}
A
B
are perpendicular.
geometry