
circle/ellipse question
Hello everybody, this is my first post here, I am no mathematician, although I have a craftsman's grasp of math inasmuch as it relates to my work as a carpenter and cabinetmaker.
Sometimes, however, I am out of my depth, and this is one such occasion, so I humbly plead for your help.
Imagine , if you will, an equilateral triangle with a long base and very shallow angles. For the sake of argument, let the base be 100 units long, and let the distance from the base line to the apex of the triangle be 10 units.
Now, I do know that there is a unique circle which will be tangent to the three vertices of the triangle.
I do believe that there is an online program to solve this problem, ie to give the radius of the circle which satisfies these conditions, but I seem to have lost the link, so question one is, can anybody give me the link to solve this problem ?
Question two is , for me, slightly more advanced (although probably totally elementary to y'all !),
I believe intuitively that for the same set of three points , there is also a unique ellipse which will be tangent to these three points, with the center point of the major axis being vertically in line with the apex of the triangle.
So question number two is , is there an online program which would tell me the ellipse which would satisfy these three points ? (ie give me the major and minor axes )
If there isn't such an online program, then can anybody point me in the direction of instructions as to how one might calculate what such an ellipse might be ?
Thanks in advance for any suggestions which might be forthcoming.

If you take the center of the base as the origin (0, 0), then the coordinates of the three points will be (50, 0), (50, 0) and (0, 10).
The general equation equation of the circle is
$\displaystyle x^2 + y^2 + 2gx + 2fy + c = 0$
Similarly the equation of the ellipse is
$\displaystyle \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$
Substitute three points in the above equations and solve to find the unknowns.