$ABCD$ is a trapezoid of a fixed big base $[AB]$ and a small base $[CD]$ of constant length. Find the sets of variable Points C & D such that the sum of $AD + BC$ remains equal to a constant length $l$
I tried using midsegment theory so I can relate [AD] and [BC] for having their sum constant but I reached a deadend or going around a circle!
First, the triangles $ICD$ and $IAC$ are similar.
$$\frac{t}{b}=\frac{p+x}{p}=\frac{q+y}{q}$$
That give us $x=\frac{p(t-b)}{b}$ and $y=\frac{q(t-b)}{b}$, but $x+y=l$, so
$$l=\frac{p(t-b)}{b}+\frac{q(t-b)}{b}=\frac{(p+q)(t-b)}{b} \Rightarrow p+q=\frac{lb}{t-b}$$
And also
$$p+q+x+y=\frac{lb}{t-b}+l=\frac{lt}{t-b}$$
So, we have that $ID+IC=p+q$ is constant and $IA+IB=p+q+x+y$ is also constant.
$I$ belongs to two ellipse, one is stopped (with focus $A$ and $B$) and the other is moving (with focus $C$ and $D$). Those are two intern tangent ellipses intersect at $I$.
So the locus of $C$ and $D$ are the focus of a variable ellipse which is tangent to a stopped ellipse with focus $A$ and $B$.