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National and Regional Contests
India Contests
India STEMS
2024 India STEMS
STEMS 2024 Math Cat A
P6
P6
Part of
STEMS 2024 Math Cat A
Problems
(1)
Orthocentre,mixed with median and symmedian
Source: STEMS 2024 CAT A P5,CAT B P4
12/17/2023
Let ABC with orthocenter
H
H
H
and circumcenter
O
O
O
be an acute scalene triangle satisfying
A
B
=
A
M
AB = AM
A
B
=
A
M
where
M
M
M
is the midpoint of
B
C
BC
BC
. Suppose
Q
Q
Q
and
K
K
K
are points on
(
A
B
C
)
(ABC)
(
A
BC
)
distinct from A satisfying
∠
A
Q
H
=
90
\angle AQH = 90
∠
A
Q
H
=
90
and
∠
B
A
K
=
∠
C
A
M
\angle BAK = \angle CAM
∠
B
A
K
=
∠
C
A
M
. Let
N
N
N
be the midpoint of
A
H
AH
A
H
. • Let
I
I
I
be the intersection of
B
-midline
B\text{-midline}
B
-midline
and
A
-altitude
A\text{-altitude}
A
-altitude
Prove that
I
N
=
I
O
IN = IO
I
N
=
I
O
. • Prove that there is point
P
P
P
on the symmedian lying on circle with center
B
B
B
and radius
B
M
BM
BM
such that
(
A
P
N
)
(APN)
(
A
PN
)
is tangent to
A
B
AB
A
B
.Proposed by Krutarth Shah
geometry