How do I find the area of the inscribed triangle within the ellipse without analytic geometry? I have the following problem where I need to solve for the inradius of a particular inscribed triangle in an ellipse:

∆ABC is situated within an ellipse whose major axis is of length 10 and whose minor axis is of length 8. Point A is a focus of the ellipse, point B is an endpoint of the minor axis and point C is on the ellipse such that the other focus lies on BC. Compute the inradius of ∆ABC. Hint: recall the area formula for a triangle involving the inradius.

At the end I'm given a hint to use the area formula for a triangle $K = rs$  in solving for the inradius. I divided both sides by $s$ to get $K/s = r$, and then tried to solve for both $K$ and $s$. It was easy to solve for $s$, because if we let $A'$ be the other focus then
$$
\begin{equation}
\begin{split}
\begin{gathered}
P = AB + BC + AC = AB + BA' + A'C + CA\\
AB = a\\
BA' = a\\
A'C + CA = 2a\\
P/2 = 2a
\end{gathered}
\end{split}
\end{equation}
$$
However, I'm having trouble solving for $K$. I realize it's easy to just write up an analytic equation for the ellipse and find the equation of the line $BA'$ then solve for $C$, but I'm interested if there's a better way to do this. I noticed that since B connects line segments passing through both foci the question may have something to do with the reflective property of the ellipse, but I'm not sure. How do I find the area of the triangle non-analytically?

*

*You know the angle(ABC) is 2t, where cos(t) = b/a.

*You know the length(BC) is a + x2a, and the length(AC) is (1-x)2a, where x is unknown between 0 and 1.

*Define two equations for Area:


*

*One equation is the side-angle-side equation: Area = length(AB)length(BC)sin(angle(ABC)/2

*One equation is Heron's Formula: Area = sqrt(P(P-length(AB)(P-length(BC)(P-length(AC))


*Now you have two simultaneous equations with two unknowns (x, Area), to plug into your favorite symbolic solver. 

