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Contests
National and Regional Contests
Latvia Contests
Latvia BW TST
2015 Latvia Baltic Way TST
7
7
Part of
2015 Latvia Baltic Way TST
Problems
(1)
O_1O_2 __|_ AB, starting with intersecting circles
Source: 2015 Latvia BW TST P7
12/16/2022
Two circle
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
intersect at points
A
A
A
and
B
B
B
, point
P
P
P
is not on the line
A
B
AB
A
B
. Line
A
P
AP
A
P
intersects again
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
at points
K
K
K
and
L
L
L
respectively, line
B
P
BP
BP
intersects again
Γ
1
\Gamma_1
Γ
1
and
Γ
2
\Gamma_2
Γ
2
at points
M
M
M
and
N
N
N
respectively and all the points mentioned so far are different. The centers of the circles circumscribed around the triangles
K
M
P
KMP
K
MP
and
L
N
P
LNP
L
NP
are
O
1
O_1
O
1
and
O
2
O_2
O
2
respectively. Prove that
O
1
O
2
O_1O_2
O
1
O
2
is perpendicular to
A
B
AB
A
B
.
perpendicular
geometry