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Sharygin Geometry Olympiad
2023 Sharygin Geometry Olympiad
5
5
Part of
2023 Sharygin Geometry Olympiad
Problems
(1)
prove perpendicularity with equal lengths and midpoint
Source: Sharygin 2023 - P5 (Grade-8)
3/4/2023
Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral. Points
E
E
E
and
F
F
F
lie on the sides
A
D
AD
A
D
and
C
D
CD
C
D
in such a way that
A
E
=
B
C
AE = BC
A
E
=
BC
and
A
B
=
C
F
AB = CF
A
B
=
CF
. Let
M
M
M
be the midpoint of
E
F
EF
EF
. Prove that
∠
A
M
C
=
9
0
∘
\angle AMC = 90^{\circ}
∠
A
MC
=
9
0
∘
.
geometry
perpendicular
Sharygin Geometry Olympiad
Sharygin 2023