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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.

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 $\triangle ABC$?

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  • $\begingroup$ Do you mean that the tangent lines will not meet ONLY in the case of right triangles? This is nonsense. (In hyperbolic geometry) $\endgroup$ – zoli May 14 '15 at 7:50
  • $\begingroup$ maybe i am wrong can you give me an example of another triangle where the "circumscribed triangle" doesn't exist? (ps the construction is in Euclidean geometry) (ps added bit to question about your question) $\endgroup$ – Willemien May 14 '15 at 7:55
  • $\begingroup$ ![enter image description here][1] If the triangle is large enough then the tangent lines hardly meet. [1]: i.stack.imgur.com/h7It7.png $\endgroup$ – zoli May 14 '15 at 8:03
  • $\begingroup$ Oh, so you consider "beyond the absolute" constructions. $\endgroup$ – zoli May 14 '15 at 8:20
  • $\begingroup$ @Willemien, suggest you get an inexpensive one-page scanner such as I use. It can make color or black and white, pdf or jpeg; jpegs are better for MSE, no idea why. In math.stackexchange.com/questions/173016/… my first answer was given diagrams by an SE moderator; later I bought the scanner and was able to upload my own diagrams as a second answer. I still use it sometimes; you, of course, have maintained an interest in geometry throughout, and would be able to use it for both questions and answers, your own drawings. Will. $\endgroup$ – Will Jagy May 14 '15 at 19:14
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What is triangle $\triangle ABC$ called in relation to triangle $\triangle DEF$?

According to Wikipedia, the inner triangle is called the Gergonne triangle, contact triangle or intouch triangle of the outer.

What is triangle $\triangle DEF$ called in relation to triangle $\triangle ABC$?

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.

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  • $\begingroup$ thanks made me wonder does the location of the Gergonne point have any special meaning in relation to $\triangle ABC$ when we see it as an hyperbolic ideal triangle? $\endgroup$ – Willemien May 15 '15 at 7:05
  • $\begingroup$ @Willemien: hyperbolically speaking, the Gergonne point would be the orthocenter, since lines through the poles of the edges of $\triangle ABC$ are orthogonal to the sides of this triangle. $\endgroup$ – MvG May 15 '15 at 7:40
  • $\begingroup$ $\triangle DEF$ is called the tangential triangle of $\triangle ABC$; see en.wikipedia.org/wiki/Tangential_triangle . $\endgroup$ – darij grinberg Aug 10 '15 at 11:21

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