What other special lines are associated to triangles? Maybe we can give them a spin . . .
The Tritangent Line
We know that any triangle ABC can in inscribed in its circumcicle. At each vertex of ABC we can draw the line tangent to the circle. This tangent line will meet the line containing the opposite side of the triangle, and so we can generate three points of intersection as there are three vertices. Here's a sketch illustrating this construction. What do you notice about these three points?
As C runs around the circumcircle, segment AB is fixed,
and the evolute of the line congruence appears to be a conic section,
either an ellipse or hyperbola depending on whether the length of
AB is greater or less than 
where r is the
radius of the circle. The evolute appears to be tangent the the
circle at points A and B. It would be interesting to
find the point on L which traces out the evolute and see if it
has any special relationship to triangle ABC.
The Orthic Axis
Further Explorations
Heidi Burgiel and Alan McRae