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Israel Olympic Revenge
2020 Israel Olympic Revenge
P4
P4
Part of
2020 Israel Olympic Revenge
Problems
(1)
Many Feuerbach points and similar quads
Source: Israel 2020 Olympic Revenge
3/22/2021
Original post by shalomrav, but for some reason the mods locked the problem without any solves :noo:Let
A
B
C
D
ABCD
A
BC
D
be a cyclic quadrilateral inscribed in circle
Ω
\Omega
Ω
. Let
F
A
F_A
F
A
be the (associated with
Ω
\Omega
Ω
) Feuerbach point of the triangle formed by the tangents to
Ω
\Omega
Ω
at
B
,
C
,
D
B,C,D
B
,
C
,
D
, that is, the point of tangency of
Ω
\Omega
Ω
and the nine-point circle of that triangle. Define
F
B
,
F
C
,
F
D
F_B, F_C, F_D
F
B
,
F
C
,
F
D
similarly. Let
A
′
A'
A
′
be the intersection of the tangents to
Ω
\Omega
Ω
at
A
A
A
and
F
A
F_A
F
A
. Define
B
′
,
C
′
,
D
′
B', C', D'
B
′
,
C
′
,
D
′
similarly. Prove that quadrilaterals
A
B
C
D
ABCD
A
BC
D
and
A
′
B
′
C
′
D
′
A'B'C'D'
A
′
B
′
C
′
D
′
are similar
geometry
Feuerbach