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Contests
National and Regional Contests
Iran Contests
Iran Team Selection Test
2024 Iran Team Selection Test
2
2
Part of
2024 Iran Team Selection Test
Problems
(1)
A right angled triangle with AC=2AB
Source: Iran Team selection test 2024 - P2
5/19/2024
For a right angled triangle
△
A
B
C
\triangle ABC
△
A
BC
with
∠
A
=
90
\angle A=90
∠
A
=
90
we have
A
C
=
2
A
B
AC=2AB
A
C
=
2
A
B
. Point
M
M
M
is the midpoint of side
B
C
BC
BC
and
I
I
I
is incenter of triangle
△
A
B
C
\triangle ABC
△
A
BC
. The line passing trough
M
M
M
and perpendicular to
B
I
BI
B
I
intersect with lines
B
I
BI
B
I
and
A
C
AC
A
C
at points
H
H
H
and
K
K
K
respectively. If the semi-line
I
K
IK
I
K
cuts circumcircle of triangle
△
A
B
C
\triangle ABC
△
A
BC
at
F
F
F
and
S
S
S
be the second intersection point of line
F
H
FH
F
H
with circumcircle of triangle
△
A
B
C
\triangle ABC
△
A
BC
, then prove that
S
M
SM
SM
is tangent to the incircle of triangle
△
A
B
C
\triangle ABC
△
A
BC
.Proposed by Mahdi Etesami Fard
geometry