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Ireland National Math Olympiad
2018 Irish Math Olympiad
5
5
Part of
2018 Irish Math Olympiad
Problems
(1)
chord of a circle tangent to another circle
Source: Irmo 2018 p1 q5
9/16/2018
Points
A
,
B
A, B
A
,
B
and
P
P
P
lie on the circumference of a circle
Ω
1
\Omega_1
Ω
1
such that
∠
A
P
B
\angle APB
∠
A
PB
is an obtuse angle. Let
Q
Q
Q
be the foot of the perpendicular from
P
P
P
on
A
B
AB
A
B
. A second circle
Ω
2
\Omega_2
Ω
2
is drawn with centre
P
P
P
and radius
P
Q
PQ
PQ
. The tangents from
A
A
A
and
B
B
B
to
Ω
2
\Omega_2
Ω
2
intersect
Ω
1
\Omega_1
Ω
1
at
F
F
F
and
H
H
H
respectively. Prove that
F
H
FH
F
H
is tangent to
Ω
2
\Omega_2
Ω
2
.
geometry
tangent
circles