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National and Regional Contests
Iran Contests
Iran MO (3rd Round)
2024 Iran MO (3rd Round)
5
5
Part of
2024 Iran MO (3rd Round)
Problems
(1)
Tangent circles in a parallelogram configuration
Source: Iran MO 3rd Round 2024 P5
8/28/2024
Let
A
B
C
D
ABCD
A
BC
D
be a parallelogram and let
A
X
AX
A
X
and
A
Y
AY
A
Y
be the altitudes from
A
A
A
to
C
B
,
C
D
CB, CD
CB
,
C
D
, respectively. A line
ℓ
⊥
X
Y
\ell \perp XY
ℓ
⊥
X
Y
bisects
A
X
AX
A
X
and meets
A
B
,
B
C
AB, BC
A
B
,
BC
at
K
,
L
K, L
K
,
L
. Similarly, a line
d
⊥
X
Y
d \perp XY
d
⊥
X
Y
bisects
A
Y
AY
A
Y
and meets
D
A
,
D
C
DA, DC
D
A
,
D
C
at
P
,
Q
P, Q
P
,
Q
. Show that the circumcircles of
△
B
K
L
\triangle BKL
△
B
K
L
and
△
D
P
Q
\triangle DPQ
△
D
PQ
are tangent to each other.
geometry