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.
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 $(0,0), (1,0), (x,y)$.
On
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.
Turns out it's the symmedian point, X6.