Let 
be a triangle. Construct two triangle 
, 
on the two sides 
. Let 
on the plane, 
is on .. Construct two lines through , and these two lines meet respectively at
, such that angle of three line are fixed when moved on the line . Denote 
respectively are on 
such that 
and 
. Prove that: When moved on the line , the circle 
through fixed point.
Solution by: Luis González:
Since
and 
have constant directions, then clearly 
is a perspectivity from 
to 
and 
is a perspectivity from to 
Analogously, 
is a perspectivity from to 
and 
is a perspectivity from to 
is a homography from to 
mapping the infinite point of into the infinite point of when is at infinity 
envelopes a fixed parabola 
tangent to circles 
go through the focus of 
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