Dynamics of ellipses inscribed in triangles

Suppose that we are given two distinct points, $P_1$ and $P_2$, in the interior of a triangle, $T$. Is there always an ellipse inscribed in $T$ which also passes through $P_1$ and $P_2$ ? If yes, how many such ellipses ? We answer those questions in this paper. It turns out that, except for $P_1$ and $P_2$ on a union of three line segments, there are four such ellipses which pass through $P_1$ and $P_2$. We also answer a similar question if instead $P_1$ and $P_2$ lie on the boundary of $T$. Finally, an interesting related question, is the following: Given a point, $P$, in the interior of a triangle, $T$, and a real number, $r$, is there always an ellipse inscribed in $T$ which passes through $P$ and has slope $r$ at $P$ ? Again, if yes, how many such ellipses ? The answer is somewhat different than for the two point case without specifying a slope. There are cases where no such ellipse exists.