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KoMaL A Problems
KoMaL A Problems 2019/2020
A. 758
A. 758
Part of
KoMaL A Problems 2019/2020
Problems
(1)
Feuerbach Circle
Source: KöMaL A. 758
3/20/2022
In a quadrilateral
A
B
C
D
,
ABCD,
A
BC
D
,
A
B
=
B
C
=
D
A
/
2
,
AB=BC=DA/\sqrt{2},
A
B
=
BC
=
D
A
/
2
,
and
∠
A
B
C
\angle ABC
∠
A
BC
is a right angle. The midpoint of
B
C
BC
BC
is
E
,
E,
E
,
the orthogonal projection of
C
C
C
on
A
D
AD
A
D
is
F
,
F,
F
,
and the orthogonal projection of
B
B
B
on
C
D
CD
C
D
is
G
.
G.
G
.
The second intersection point of circle
(
B
C
F
)
(BCF)
(
BCF
)
(with center
H
H
H
) and line
B
G
BG
BG
is
K
,
K,
K
,
and the second intersection point of circle
(
B
C
F
)
(BCF)
(
BCF
)
and line
H
K
HK
HK
is
L
.
L.
L
.
The intersection of lines
B
L
BL
B
L
and
C
F
CF
CF
is
M
.
M.
M
.
The center of the Feurbach circle of triangle
B
F
M
BFM
BFM
is
N
.
N.
N
.
Prove that
∠
B
N
E
\angle BNE
∠
BNE
is a right angle.Proposed by Zsombor Fehér, Cambridge
geometry
komal
Feuerbach