# Equilic Quadrilateral II: What is this about?

A Mathematical Droodle

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Copyright © 1996-2018 Alexander BogomolnyThe applet may suggest the following generalization of a theorem from [Honsberger] concerning *equilic* quadrilaterals:

Assume in a quadrilateral ABCD

(In [Honsberger] it is stated that, equilic ABCD, ΔKLN is equilateral. That ΔKMN is also quadrilateral is shown similarly. Putting the two together, KMLN is seen to be a rhombus. However, for it to be a rhombus, it is not necessary that the angle between AD and BC be 60°.)

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The sides of the quadrilateral KMLN being the midlines in two pairs of triangles, one with base CD, the other base BC, the four sides are necessarily equal; hence also are pairwise parallel.

However, in an equilic quadrilateral, rhombus KLMN consists of two equilateral triangles.

## References

- R. Honsberger,
*Mathematical Gems III*, MAA, 1985, pp. 32-35

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Copyright © 1996-2018 Alexander Bogomolny