@vv Maple's own solve command produces a correct inellipse within about one second.

@vv Thank you, vv. Does this mean that an inellipse exists for any two points of tangency, each on a different side, in any shape of triangle?

@Pepini You might be interested in this.

Contour_colored_bands.mw

@Kitonum Thank you for advancing my understanding.

@acer Thanks for this info, I didn't realize the effect abs would have on the plotted expression.

Does Maple Learn have a 3D capability?

@Carl Love Just before I uploaded my worksheet I made a last second change that omitted a colon. This is fixed in the worksheet below. Interestingly this error only caused a "Cannot parse..." error message on my machine under Maple 2020.

I just executed this worksheet and it consumed just under one gigabyte and 230 seconds. I hope it now works for you.

I'm afraid I could not understand how to apply the color technique you kindly explained to individual cells, so I tried one I was familiar with.

Stereographic_projection_with_colors.mw

@Carl Love Here is an effort at displaying stereographic projections of a grid on a sphere onto the plane on which the sphere rests, colored in three ways: a gradient of color, colored bands, and individual colored cells.

The worksheet defines two new coordinate systems, as you have suggested.

One of them is used to "reverse" project a grid on the plane onto the sphere.

The other projects the uncolored grid from the sphere to the plane.

However I could not find a way of projecting a colored grid from the sphere to the plane, so the colors are then applied directly to the plane grid.

Stereographic_projection_with_colors.mw

@acer Thanks for this very useful information

@Carl Love You have provided a more universally applicable answer than I expected and it will take me some time and effort to digest.

I am grateful to you for the time and effort which you have put into this.

@acer I was inspired to do this animation by viewing 

https://www.youtube.com/watch?v=0z1fIsUNhO4

This youtube presentation shows a gradient style coloring of the cells which I prefer.

However, if you have any other coloring suggestions I would be pleased to see them.

This is an example of shading between spacecurves. One of the spacecurves can be a single point. For 2D set the z spacecurve component to zero or transform 3D to 2D.

eq1 := <0.8000*cos(t), 0, sin(t)>;
eq2 := <2*cos(t), sin(t), 3>;
C1 := plots[spacecurve](eq1, t = 0 .. 2*Pi, color = blue, thickness = 3);
C2 := plots[spacecurve](eq2, t = 0 .. 2*Pi, color = blue, thickness = 3);
S := plot3d(s*eq1 + (1 - s)*eq2, t = 0 .. 2*Pi, s = 0 .. 1, style = surface, color = yellow);
display([S, C1, C2], scaling = constrained, axes = boxed, axis[1] = [color = red], axis[2] = [color = blue], axis[3] = [color = green]);

@acer This looks like just what I was after. I thank you.

@Carl Love After much effort I failed to find documentation describing your ingeneous use of this restriction on parameter fnc.

Please tell me where Maple help explains this.

@Rouben Rostamian I appreciate the head start you've given me. I'll work on it. Much thanks