This is similar to some questions that have been asked (e.g. construct-triangle-given-inradius-and-circumradius), but I don't see the exact same question. It arose out of an inversive geometry formula I was working with: For any non-obtuse triangle $\Delta ABC$, the inversive distance $\delta$ of the circumcircle and 9-point circle is $acosh(2 \cdot (cos(\angle A) \cdot cos(\angle B) \cdot cos(\angle C)) + 1)$, where inversive distance is defined to be the log of the ratio of radii of two concentric circles into which the 9-point and circumcircles can be inverted. In the diagram the value of $\delta$ shown is calculated directly from the radii of the concentric circles, but if I take my calculator and calculate directly $acosh(2 \cdot (cos(\angle A) \cdot cos(\angle B) \cdot cos(\angle C)) + 1)$ using the value shown as calculated from $\Delta ABC$, I get the same value as expected.
I got to this question by first inverting into the concentric circles, verifying the formula, and then reinverting somewhat arbitrarily into two other circles. According to the formula, the two new circles are the 9-point and circumcircle of a triangle similar to the original. I'd prefer not to have to do something like send points from the original triangle over via the same two inversions (if this would even work as expected ... I'm not sure about that). E.g. I'd like to be able to fill in the new similar triangle based only on the 9-point and circumcircle. The scenario (and illustration of the formula) is shown in the diagram below, with the two circles in the upper left the "new" circum and 9-point circles that I want the triangle for (the three collinear points the circumcenter, 9-point center, and orthocenter).
As always, I have a feeling I'm missing the obvious, but then again maybe not!
