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.

Let's assume you could divide everything into quadrilaterals. Now, let's count the number of edges in this division; call this number $E$ (in your example, I count $E=15$, but I might have miscounted). Now, each quadrilateral has 4 edges, but with the exception of the 3 edges on the border of the triangle, each edge belongs to 2 quadrilaterals. Therefore, if there are $Q$ quadrilaterals, we have that $4Q = 2E-3$.
But the left side of this equation is even and the right side is odd, leading to a contradiction.
In general, a similar argument will show that you always need an even number of odd-sided faces in any planar graph. 
