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Any scalene triangle can be dissected into 4 similar but non-congruent triangles in three ways, each with a single pair of congruent triangles. Lines connecting the opposing vertices of these congruent triangles happen to concur at a point.

similar triangle point

Which triangle center is this?

The point is at $\left\{\frac{x^2+x+y^2}{2 \left((x-1) x+y^2+1\right)},\frac{y}{2 \left((x-1) x+y^2+1\right)}\right\}$ for triangle $(0,0), (1,0), (x,y)$.

Ed Pegg
  • 20,955

2 Answers2

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Turns out it's the symmedian point, X6.

Ed Pegg
  • 20,955
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I have just released a new webpage which makes it much easier to find these centres given a triangle and its cartesian, trilinear or barycentric coordinates:

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Triangle/tricoords.html

It relies on Clark Kimberling's Encyclopedia for Triangle Centers (ETC) at http://faculty.evansville.edu/ck6/encyclopedia/ETC.html

It may have a glitch or two so please send comments and corrections to the email address on the page.

Ron Knott
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  • These links might give an answer to this question, but it is better to include the essential parts of the answer in the post itself. That way, the answer is always available, even in case the links vanish. – Ernie060 Mar 29 '19 at 10:27