Let ABC be a triangle with interior angles $\alpha, \beta$ and $\gamma.$ If this triangle inscribed in an ellipse with semi-major axis $a$ and semi-minor axis $b,$ how can we find the radius of the circumcircle of ABC?
I am thinking of this problem for a long time, but could not think of a way to solve it. Any idea?
It is simple: you cannot solve it, because the given parameters do not fix $R$. Given three non-collinear points in the plane, here it is a set of ellipses with aspect ratio $2:1$ going through such points:
This can be drawn by picking a direction, scaling with respect to the orthogonal direction, drawing the circumcircle of the scaled triangle and scaling it back to have an ellipse with aspect ratio $2:1$ going through the vertices of the original triangle. In particular, $\alpha,\beta,\gamma,a,b$ do not fix $R$. The next animation is simply derived from the previous one: