Equilic Quadrilateral

A Quadrilateral in which a pair of opposite sides have the same length and are inclined at 60° to each other (or equivalently, satisfy $\left\langle{A}\right\rangle{}+\left\langle{B}\right\rangle{}=120^\circ$). Some interesting theorems hold for such quadrilaterals. Let $ABCD$ be an equilic quadrilateral with $AD=BC$ and $\left\langle{A}\right\rangle{}+\left\langle{B}\right\rangle{}=120^\circ$. Then

1. The Midpoints $P$, $Q$, and $R$ of the diagonals and the side $CD$ always determine an Equilateral Triangle.

2. If Equilateral Triangle $PCD$ is drawn outwardly on $CD$, then $\Delta PAB$ is also an Equilateral Triangle.

3. If Equilateral Triangles are drawn on $AC$, $DC$, and $DB$ away from $AB$, then the three new Vertices $P$, $Q$, and $R$ are Collinear.

See Honsberger (1985) for additional theorems.

References

Garfunkel, J. ``The Equilic Quadrilateral.'' Pi Mu Epsilon J. 7, 317-329, 1981.

Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 32-35, 1985.