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National and Regional Contests
Croatia Contests
Croatia MO (HMO) - geometry
2020.7
2020.7
Part of
Croatia MO (HMO) - geometry
Problems
(1)
perpendicular wanted, AR=RQ, <CAP =<ABC, circle of diameter
Source: 2020 Croatia MO p7
2/16/2021
A circle of diameter
A
B
AB
A
B
is given. There are points
C
C
C
and
D
D
D
on this circle, on different sides of the diameter such that holds
A
C
<
B
C
AC <BC
A
C
<
BC
or
A
C
<
A
D
AC<AD
A
C
<
A
D
. The point
P
P
P
lies on the segment
B
C
BC
BC
and
∠
C
A
P
=
∠
A
B
C
\angle CAP = \angle ABC
∠
C
A
P
=
∠
A
BC
. The perpendicular from the point
C
C
C
to the line
A
B
AB
A
B
intersects the direction
B
D
BD
B
D
at the point
Q
Q
Q
. Lines
P
Q
PQ
PQ
and
A
D
AD
A
D
intersect at point
R
R
R
, and the lines
P
Q
PQ
PQ
and
C
D
CD
C
D
intersect at point
T
T
T
. If
A
R
=
R
Q
AR=RQ
A
R
=
RQ
, prove that the lines
A
T
AT
A
T
and
P
Q
PQ
PQ
are perpendicular.
geometry
perpendicular