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Iran Team Selection Test
2024 Iran Team Selection Test
7
7
Part of
2024 Iran Team Selection Test
Problems
(1)
Not a nice geo , so well ...
Source: Iran Team selection test 2024 - P7
5/19/2024
Let
△
A
B
C
\triangle ABC
△
A
BC
and
△
C
′
B
′
A
\triangle C'B'A
△
C
′
B
′
A
be two congruent triangles ( with this order and orient. ). Define point
M
M
M
as the midpoint of segment
A
B
AB
A
B
and suppose that the extension of
C
B
′
CB'
C
B
′
from
B
′
B'
B
′
passes trough
M
M
M
, if
F
F
F
be a point on the smaller arc
M
C
MC
MC
of circumcircle of triangle
△
B
M
C
\triangle BMC
△
BMC
such that
∠
F
B
′
A
=
90
\angle FB'A=90
∠
F
B
′
A
=
90
and
∠
C
′
C
B
′
≠
90
\angle C'CB' \neq 90
∠
C
′
C
B
′
=
90
, then prove that
∠
B
′
C
′
C
=
∠
C
A
F
\angle B'C'C=\angle CAF
∠
B
′
C
′
C
=
∠
C
A
F
.Proposed by Alireza Dadgarnia
geometry