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Turkey Contests
Turkey Team Selection Test
2018 Turkey Team Selection Test
9
9
Part of
2018 Turkey Team Selection Test
Problems
(1)
Set of Simson Lines
Source: Turkey Team Selection Test 2018 P9
3/27/2018
For a triangle
T
T
T
and a line
d
d
d
, if the feet of perpendicular lines from a point in the plane to the edges of
T
T
T
all lie on
d
d
d
, say
d
d
d
focuses
T
T
T
. If the set of lines focusing
T
1
T_1
T
1
and the set of lines focusing
T
2
T_2
T
2
are the same, say
T
1
T_1
T
1
and
T
2
T_2
T
2
are equivalent. Prove that, for any triangle in the plane, there exists exactly one equilateral triangle which is equivalent to it.
geometry
Simson Lines