[Math] How to build a trapezoid

Question

Build a trapezoid knowing its diagonals, the angle between them, and, also, the sum of $2$ adiacent sides.

I appreciate your time and help!

Answer 1

Given the lengths $\ell_1, \ell_2$ of two diagonals and the angle $\alpha$ between them, let $2a$ be the required sum of lengths of two adjacent sides.

1. Construct three helper points $A, B, C$ such that $AB = \ell_1, AC = \ell_2$ and $\measuredangle BAC = \alpha$.
   ($AB$ and $AC$ are the two blue line segments in diagram below)
2. Construct an ellipse having $A, B$ as foci with semi-major axis $a$.
   (The red ellipse in diagram below)
3. Start from the two line segments $AB$ and $AC$, construct parallelogram $ABEC$ and let the line $AE$ intersect the ellipse at $F$.
4. Start from the line segments $AC$ and $CF$, construct parallelogram $AGFC$.

Take $AB$ and $GF$ as the two desired diagonal. The quadrilateral $AGBF$ will be a trapezoid one seek. To see this.

• By construction, $|AB| = \ell_1$. Since $AGFC$ is a parallelogram, $|GF| = |AC| = \ell_2$ and angle between $AB$ and $GF$ is the angle $\measuredangle BAC = \alpha$.
• Since $F$ lies on the ellipse having $A$, $B$ as foci. $|AF| + |BF| = 2a$.
• Since $ABEC$ and $AGFC$ are parallelograms, $GBEF$ is a parallelogram too. As a result, $GB \parallel FE \parallel AF$.