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JOM 2015 Shortlist
G7
G7
Part of
JOM 2015 Shortlist
Problems
(1)
Prove that there exists $6$ pairs of tangent circles
Source: Junior Olympiad of Malaysia Shortlist 2015 G7
7/17/2015
Let
A
B
C
ABC
A
BC
be an acute triangle. Let
H
A
,
H
B
,
H
C
H_A,H_B,H_C
H
A
,
H
B
,
H
C
be points on
B
C
,
A
C
,
A
B
BC,AC,AB
BC
,
A
C
,
A
B
respectively such that
A
H
A
⊥
B
C
,
B
H
B
⊥
A
C
,
C
H
C
⊥
A
B
AH_A\perp BC, BH_B\perp AC, CH_C\perp AB
A
H
A
⊥
BC
,
B
H
B
⊥
A
C
,
C
H
C
⊥
A
B
. Let the circumcircles
A
H
B
H
C
,
B
H
A
H
C
,
C
H
A
H
B
AH_BH_C,BH_AH_C,CH_AH_B
A
H
B
H
C
,
B
H
A
H
C
,
C
H
A
H
B
be
ω
A
,
ω
B
,
ω
C
\omega_A,\omega_B,\omega_C
ω
A
,
ω
B
,
ω
C
with circumcenters
O
A
,
O
B
,
O
C
O_A,O_B,O_C
O
A
,
O
B
,
O
C
respectively and define
O
A
B
∩
ω
B
=
P
A
B
≠
B
O_AB\cap \omega_B=P_{AB}\neq B
O
A
B
∩
ω
B
=
P
A
B
=
B
. Define
P
A
C
,
P
B
A
,
P
B
C
,
P
C
A
,
P
C
B
P_{AC},P_{BA},P_{BC},P_{CA},P_{CB}
P
A
C
,
P
B
A
,
P
BC
,
P
C
A
,
P
CB
similarly. Define circles
ω
A
B
,
ω
A
C
\omega_{AB},\omega_{AC}
ω
A
B
,
ω
A
C
to be
O
A
P
A
B
H
C
,
O
A
P
A
C
H
B
O_AP_{AB}H_C,O_AP_{AC}H_B
O
A
P
A
B
H
C
,
O
A
P
A
C
H
B
respectively. Define circles
ω
B
A
,
ω
B
C
,
ω
C
A
,
ω
C
B
\omega_{BA},\omega_{BC},\omega_{CA},\omega_{CB}
ω
B
A
,
ω
BC
,
ω
C
A
,
ω
CB
similarly.Prove that there are
6
6
6
pairs of tangent circles in the
6
6
6
circles of the form
ω
x
y
\omega_{xy}
ω
x
y
.
geometry