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Mathley Magazine
Geometry Mathley 2011-12
12.4
12.4
Part of
Geometry Mathley 2011-12
Problems
(1)
Geometry Mathley 12.4 orthogonal circles pairwise
Source:
6/7/2020
Quadrilateral
A
B
C
D
ABCD
A
BC
D
has two diagonals
A
C
,
B
D
AC,BD
A
C
,
B
D
that are mutually perpendicular. Let
M
M
M
be the Miquel point of the complete quadrilateral formed by lines
A
B
,
B
C
,
C
D
,
D
A
AB,BC,CD,DA
A
B
,
BC
,
C
D
,
D
A
. Suppose that
L
L
L
is the intersection of two circles
(
M
A
C
)
(MAC)
(
M
A
C
)
and
(
M
B
D
)
(MBD)
(
MB
D
)
. Prove that the circumcenters of triangles
L
A
B
,
L
B
C
,
L
C
D
,
L
D
A
LAB,LBC,LCD,LDA
L
A
B
,
L
BC
,
L
C
D
,
L
D
A
are on the same circle called
ω
\omega
ω
and that three circles
(
M
A
C
)
,
(
M
B
D
)
,
ω
(MAC), (MBD), \omega
(
M
A
C
)
,
(
MB
D
)
,
ω
are pairwise orthogonal.Nguyễn Văn Linh
circles
Concyclic