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Tournament Of Towns
1992 Tournament Of Towns
(327) 4
(327) 4
Part of
1992 Tournament Of Towns
Problems
(1)
3 lines concurrent with a circle
Source: TOT 327 1992 Spring Α J4 - Tournament of Towns
6/9/2024
Let
P
P
P
be a point on the circumcircle of triangle
A
B
C
ABC
A
BC
. Construct an arbitrary triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
whose sides
A
1
B
1
A_1B_1
A
1
B
1
,
B
1
C
1
B_1C_1
B
1
C
1
and
C
1
A
1
C_1A_1
C
1
A
1
are parallel to the segments
P
C
PC
PC
,
P
A
PA
P
A
and
P
B
PB
PB
respectively and draw lines through the vertices
A
1
A_1
A
1
,
B
1
B_1
B
1
and
C
1
C_1
C
1
and parallel to the sides
B
C
BC
BC
,
C
A
CA
C
A
and
A
B
AB
A
B
respectively. Prove that these three lines have a common point lying on the circumcircle of triangle
A
1
B
1
C
1
A_1B_1C_1
A
1
B
1
C
1
.(V. Prasolov)
geometry
concurrency