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National and Regional Contests
Switzerland Contests
Switzerland Team Selection Test
2023 Switzerland Team Selection Test
4
4
Part of
2023 Switzerland Team Selection Test
Problems
(1)
Similar triangles with a common vertex
Source: Swiss TST 2023 P4
9/8/2023
Let
A
B
C
ABC
A
BC
and
A
M
N
AMN
A
MN
be two similar, non-overlapping triangles with the same orientation, such that
A
B
=
A
C
AB=AC
A
B
=
A
C
and
A
M
=
A
N
AM=AN
A
M
=
A
N
. Let
O
O
O
be the circumcentre of the triangle
M
A
B
MAB
M
A
B
. Prove that the points
O
,
C
,
N
O, C, N
O
,
C
,
N
and
A
A
A
lie on a circle if and only if the triangle
A
B
C
ABC
A
BC
is equilateral.
geometry