Problem: Let 
be a triangle, Let 
is the first or the secon Fermat point. Construc two line through and angle of two lie is 60 degrees. Two lines meet three sideline at 
(show in the figure attachment). Prove that 
, 
, 
are concurrent.
be a triangle, Let is the first or the secon Fermat point. Construc two line through and angle of two lie is 60 degrees. Two lines meet three sideline at (show in the figure attachment). Prove that , , are concurrent.
Solutionby Luis González:
We also show that the concurrency point D of the referred lines is on the trilinear polar of F WRT ABC.
Let
be the cevian triangle of WRT 
and 
is trilinear polar of WRT 
i.e. perspectrix of 
and 
We use polarity WRT an abitrary circle
centered at 
Polars of 
bound an equilateral triangle since they are perpendicular to 
i.e. poles 
of 
are vertices of equilateral triangle. Polars of 
are then the parallels through 
to 
respectively, meeting at the pole 
of 
If 
are the poles of 
then 
is also equilateral, being the antimedial of 
and 
are polars of 
meeting at the pole 
of 
the center of Polars of 
are two parallels through and polars of 
are two parallels through 
(these directions forming 
due to 
) poles 
of 
form parallelogram 
circumscribed to Since
then clearly the circles 
and 
meet at 
and 
which is then midpoint of their arcs 
If 
cuts again at 
we have 
are collinear their polars 
concur at 
By similar reasoning goes through 
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