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National and Regional Contests
Italy Contests
ITAMO
2021 ITAMO
5
5
Part of
2021 ITAMO
Problems
(1)
Perpendicularity and concurrence of two circles and a line
Source: ITAMO 2021 - Problem 5
5/8/2021
Let
A
B
C
ABC
A
BC
be an acute-angled triangle, let
M
M
M
be the midpoint of
B
C
BC
BC
and let
H
H
H
be the foot of the
B
B
B
-altitude. Let
Q
Q
Q
be the circumcenter of
A
B
M
ABM
A
BM
and let
X
X
X
be the intersection point between
B
H
BH
B
H
and the axis of
B
C
BC
BC
.Show that the circumcircles of the two triangles
A
C
M
ACM
A
CM
,
A
X
H
AXH
A
X
H
and the line
C
Q
CQ
CQ
pass through a same point if and only if
B
Q
BQ
BQ
is perpendicular to
C
Q
CQ
CQ
.
geometry
circumcircle