I am meeting the following triangle more and more in my investigations of ideal triangles in the Beltrami Klein model of hyperbolic geometry. That made me wonder: is there a name for it? (And does it have its own Wikipedia or Mathworld page?)
PS: the construction below is a construction in euclidean geometry (the Beltrami Klein model is a euclidean (nonconformal) representation of the complete hyperbolic plane), and in the Beltrami Klein model the triangle $\triangle ABC$ is an ideal triangle.
let $\triangle ABC$ be any triangle. (For the moment, only triangles where no angle is right should be conSidered.)
At the point $A$ draw the line $a$ tangent to the circumscribed circle.
At the point $B$ draw the line $b$ tangent to the circumscribed circle.
At the point $C$ draw the line $c$ tangent to the circumscribed circle.
Most times the lines $a,b$ and $c$ will pairwise intersect.
(Only when one of the angles is right will two of these not intersect but be parallel.)
Point $D$ is the point where $b$ and $c$ intersect.
Point $E$ is the point where $a$ and $c$ intersect.
Point $F$ is the point where $a$ and $b$ intersect.
What is triangle $\triangle ABC $ called in relation to triangle $\triangle DEF$?
Or what is triangle $\triangle DEF $ called in relation to triangle $\triangle ABC$?
According to Wikipedia, the inner triangle is called the Gergonne triangle, contact triangle or intouch triangle of the outer.
I don't know an answer to this yet, but searching the web with the three names given above might turn up some established name for the converse as well.