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Problems
Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2021 Latvia Baltic Way TST
P10
P10
Part of
2021 Latvia Baltic Way TST
Problems
(1)
Circle with lots of symmetry
Source: Latvia BW TST 2021 P10
9/26/2021
Circle
ω
\omega
ω
with centre
M
M
M
and diameter
X
Y
XY
X
Y
is given. Point
A
A
A
is chosen on
ω
\omega
ω
so that
A
X
<
A
Y
AX<AY
A
X
<
A
Y
. Points
B
,
C
B, C
B
,
C
are chosen on segments
X
M
,
Y
M
XM, YM
XM
,
Y
M
, respectively, in a way that
B
M
=
C
M
BM=CM
BM
=
CM
. A parallel line to
A
B
AB
A
B
is constructed through
C
C
C
; the line intersects
ω
\omega
ω
at
P
P
P
so that
P
P
P
lies on the smaller arc
A
Y
^
\widehat{AY}
A
Y
. Similarly, a parallel line to
A
C
AC
A
C
is constructed through
B
B
B
; the line intersects
ω
\omega
ω
at
Q
Q
Q
so that
Q
Q
Q
lies on the smaller arc
X
A
^
\widehat{XA}
X
A
. Lines
P
Q
PQ
PQ
and
X
Y
XY
X
Y
intersect at
S
S
S
. Prove that
A
S
AS
A
S
is tangent to
ω
\omega
ω
.
geometry
symmetry