Equilic Quadrilateral II: What is this about?
A Mathematical Droodle

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Discussion

The applet may suggest the following generalization of a theorem from [Honsberger] concerning equilic quadrilaterals:

Assume in a quadrilateral ABCD AD = BC, K, L, M, N are the midpoints of AC, AB, BD, AD. Then the quadrilateral KMLN is a rhombus.

(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