MathDB
Problems
Contests
National and Regional Contests
Poland Contests
Poland - First Round
1993 Poland - First Round
4
4
Part of
1993 Poland - First Round
Problems
(1)
Circles, cords, and symmetry
Source: Poland Math Olympiad 1993 Round 1 #4
6/3/2023
Given is a circle with center
O
O
O
, point
A
A
A
inside the circle and a chord
P
Q
PQ
PQ
which is not a diameter and passing through
A
A
A
. The lines
p
p
p
and
q
q
q
are tangent to the given circle at
P
P
P
and
Q
Q
Q
respectively. The line
l
l
l
passing through
A
A
A
and perpendicular to
O
A
OA
O
A
intersects the lines
p
p
p
and
q
q
q
at
K
K
K
and
L
L
L
respectively. Prove that
∣
A
K
∣
=
∣
A
L
∣
|AK| = |AL|
∣
A
K
∣
=
∣
A
L
∣
.
symmetry