Finding points for Conway Tessellation First time on Math SE, so any feedback appreciated 
I'm working on some generative art that will draw Pinwheel tiles with svgs. To that end I'm building a function that given a set of three points, spits out the seven points that define the triangle's decomposition.
For the life of me I can't remember how to find these points when the triangle isn't aligned to the x,y plane. I'd like to get the coordinates for the given point:

Is the correct approach to do some basic unit circle trig from the top left point to find the x & y distances from that point in a local coordinate system and then transform back to the coordinate system the rest of the points are in? 
Thank you. 
 A: Given three points $A, B, C \in \mathbb R^2$ such that $$(a^2, b^2, c^2) = (BC^2, CA^2, AB^2) \propto (1, 4, 5)$$ respectively, it is quite straightforward to compute the coordinates of the other vertices in your diagram.  Denote the circled point $D$, and let $E$, $F$ be the midpoints of $\overline{DC}$ and $\overline{DA}$, respectively; finally, let $G$ be the midpoint of $\overline{CA}$.  Then by similarity of triangles, $$\triangle BDC \sim \triangle BCA$$ and in particular $$\frac{BD}{BC} = \frac{BC}{AB},$$ hence $$BD = \frac{BC^2}{AB} = \frac{BC}{\sqrt{5}}.$$  That implies that $AB = 5BD$.  It follows that $$D = \frac{A + 4B}{5},$$ and the rest is straightforward:  $$E = \frac{D+C}{2} = \frac{A+4B+5C}{10}, \\ F = \frac{3A+2B}{5}, \\ G = \frac{A+C}{2}.$$
So for example, if $$A = (3 + \sqrt{3}, 3), \\ B = (5/2, 2 + \sqrt{3}/2), \\ C = (3,2),$$ then $$D = \left(\frac{(3 + \sqrt{3})+4(5/2)}{5}, \frac{3 + 4(2+\sqrt{3}/2)}{5}\right) = \left(\frac{13+\sqrt{3}}{5}, \frac{11 + 2\sqrt{3}}{5}\right).$$
