Is it possible to decompose a triangle into quads without splitting edges? By quads I mean four sided shapes. You can add vertex anywhere inside the triangle, but you can not add vertex onto existing edges, i.e., splitting them.
I tried but currently it appears to be impossible (see figure below for an attempt). If I guess correctly there must be some topological constraints that make this impossible.

Nah, it's impossible. Let's say it were possible, and you had a working dissection. Now imagine you draw four points in each quadrilateral, one on each side. Now of course the number of points is divisible by four, so it is even. On the other hand, you can pair all but three points up with the corresponding point on the other edge, so the number of points is three more than an even number, therefore it must be odd as well.
This is a contradiction, hence the dissection you are looking for is impossible.

