Find an ellipse tangent to triangle given centre position of the ellipse I have a fully defined triangle, $PT_1T_2$, of side lengths $a$, $b$, & $c$ with corresponding opposite angles $A$, $B$, & $C$. I want to find an ellipse that is tangent to the lines $PT_1$ and $PT_2$ at points $T_1$ and $T_2$ respectively.
As there are many ellipses that satisfy this condition, I impose that the centre point, $O$, of the ellipse is at a distance $d$ from point $P$ and that the line $OP$ is at an angle $D$ from the line $PT_1$, where $0<D<A$.
Given these conditions, can I find the focus of the ellipse and the angle of the major axis relative to the line $PO$? What are the equations for these?



After construction of the third tangent point $Q$
as described in the other answer,
we can consider the construction 
as the triangle $ABC$ with generalized Steiner inellipse,
for which the foci can be found as
the roots of derivative of the function
\begin{align}
p(z)&=(z-A)^u(z-B)^v(z-C)^w
,\quad u,v,w>0,\ u+v+w=1
.
\end{align} 
In this case 
\begin{align}
u&=
\frac{|AT_2|\cdot|CQ|}{|CT_2|\cdot|BC|+|AT_2|\cdot|CQ|}
,\\
v&=
\frac{|CT_2|\cdot|BQ|}{|CT_2|\cdot|BC|+|AT_2|\cdot|CQ|}
,\\
w&=1-u-v
\end{align}
and the foci are located at
\begin{align}
F_{1,2}&=
\tfrac12\,(u+v+w)^{-1}\cdot 
\Big(
(v+w)\,A+
(w+u)\,B+
(u+v)\,C
\\
&\pm
\left(
 (w+v)^2\,A^2 
+(u+w)^2\,B^2
+(u+v)^2\,C^2
\right.
\\ 
&+2\,(u\,v-v\,w-w\,u-w^2)\,A\cdot B
\\
&+2\,(v\,w-v\,u-w\,u-u^2)\,B\cdot C
\\
&+\left. 2\,(w\,u-v\,w-v\,u-v^2)\,C\cdot A
\right)^{1/2}
\Big)
.
\end{align}
