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Ireland National Math Olympiad
2015 Irish Math Olympiad
4
4
Part of
2015 Irish Math Olympiad
Problems
(1)
starting with a circle with diameter the centres of 2 externally tangent circles
Source: Irmo 2015 p1 q4
9/16/2018
Two circles
C
1
C_1
C
1
and
C
2
C_2
C
2
, with centres at
D
D
D
and
E
E
E
respectively, touch at
B
B
B
. The circle having
D
E
DE
D
E
as diameter intersects the circle
C
1
C_1
C
1
at
H
H
H
and the circle
C
2
C_2
C
2
at
K
K
K
. The points
H
H
H
and
K
K
K
both lie on the same side of the line
D
E
DE
D
E
.
H
K
HK
HK
extended in both directions meets the circle
C
1
C_1
C
1
at
L
L
L
and meets the circle
C
2
C_2
C
2
at
M
M
M
. Prove that (a)
∣
L
H
∣
=
∣
K
M
∣
|LH| = |KM|
∣
L
H
∣
=
∣
K
M
∣
(b) the line through
B
B
B
perpendicular to
D
E
DE
D
E
bisects
H
K
HK
HK
.
geometry
circles
tangent