Sage Cell Server

Orthogonal trapezoids.

A Sagecell interact to experiment with orthogonal trapezoids. \

carl.eberhart@uky.edu

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.

You can change the values in one or more inputs and press the Update button below to get the new pair of orthogonal trapezoids.
Every pair of orthogonal trapezoids is determined by the values of their 9 parameters. Of course, not every set of values yields a pair of orthogonal trapezoids. For example, if you set A1 smaller than B1, you get garbage. You may also get two trapezoids which don't intersect in a parallelogram (this will happen when a vertex of one trapezoid ends up inside the other trapezoid).