Let $\mathcal{T}=A,B,C,D$ be a trapezoid with the vertices labelled in counterclockwise order so that $A\,D$ is parallel with $B\,C$. $A\,C$ and $B\,D$ are called the first and second diagonals of $\mathcal{T}$, and $A\,B$ and $C\,D$ are called the sides of $\mathcal{T}$

An interesting theorem about trapezoids: (1). For each point $X$ on side $A\,B$, there is a unique point $Y$ on side $C\,D$ such that the area of $A\,X\,Y\,D$ is equal with the area of $A\,C\,D$.
Call $Y$ the opposite point of $X$ on $\mathcal{T}$ and label it $\bar{X}$.

A pair $\mathcal{T}_{1}=A_1,B_1,C_1,D_1$ and $\mathcal{T}_{2}=A_2,B_2,C_2,D_2$ of distinct trapezoids whose intersection is a parallelogram not containing any of the vertices is called an orthogonal pair of trapezoids provided the first and second diagonals of $\mathcal{T}_{1}$ are orthogonal respectively with the first and second diagonals of $\mathcal{T}_{2}$.

An interesting theorem about orthogonal trapezoids: (2) For each $X$ on the side $A_1\,B_1$ of $\mathcal{T}_1$ there is a unique $Z$ on the side $A_2\,B_2$ of $\mathcal{T}_{2}$ such that $X\,\bar{X}$ and $Z\,\bar{Z}$ are orthogonal. Further, as $X$ moves from $A_1$ to $B_1$ on $A_1\,B_1$, the corresponding $Z$ moves from $A_2$ to $B_2$ on $A_2\,B_2$.

A pair of orthogonal trapezoids is determined when the 4 points $A_1, C_1, A_2, C_2$ are specified together with 5 angles $\alpha_1$, $\beta_1$, $\alpha_2$, $\beta_2$, and $\theta$, where $\theta = \angle A_1Cn B_1$ where $Cn$ is the intersection of $A_1C_1$ and $A_2C_2$, and $\alpha_i= \angle C_iA_iB_i$, $\beta_i= \angle A_iC_iD_i$ for $i=1,2$. <\p>

In the interact below, press Start and you will see two rows of labeled input boxes, and after a few seconds some graphics will appear; one of two orthogonal trapezoids, the other a graph of an area function $f(s)$. When $s \in [0,1]$, $f(s)$ is the sum of the pink area and the blue area in the trapezoids.