This is a strange question, by I "feel" that the math behind it can be quite simple. It has to with constructing an ellipse (I think)!

A, B and C are angles (or points) of a triangle.

a, b and c are the respective opposing sides.

Side c and its opposing angle C are both held constant.

Now all possible triangles are formed given these restrictions. I.e. all combinations of A, B, a and b's are choosen which allow ABC to remain a triangle.

Given that the location of side c is held constant, the possible point C's will form av curve around that base c. This shape, according to my experiments, seems to be an ellips, although maybe not well defined very close to the base c.

I would like to know if there is a simple relationship between the parameters of the ellips (its two axes and its center point) and the side c and angle C of the triangle.

Code:

.C . .
. |\ .
. | \ .
. | \ .
. | \ .
. | \ .
. |_____\.
A c B

The points above is the ellips. On each such point, angle C of triangle ABC will be constant towards opposing side c. (Though this might not be correct for the points very close to base c.)

How do I determine the parameters of such an ellipse given values of C and c?