Li-Fan Chen says to YSITD
$ e_x = \frac{p_2-p_1}{|p_2-p_1|} $ $ i = e_x \cdot (p_3-p_1)$ $ a = (p_3 - p_1)-ie_x $ $ e_y = \frac{a}{|a|}$ $e_z = e_x \times e_y $ $d = |p_2-p_1|$ $ j = e_y \cdot (p_3 - p_1) $ $ x = \frac{\sqrt{r_1} - \sqrt{r_2} + \sqrt{d}}{2d}$ $ y = \frac{\sqrt{r_1} - \sqrt{r_3} + \sqrt{i} + \sqrt{j}}{2j- \frac{xi}{j}} $ $ b = \sqrt{r_1} - \sqrt{x} - \sqrt{y}$ $ a = p_1 + xe_x + ye_y $ $ p4a = a + ze_z$ $p4b = a - ze_z$