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Equilic
Quadrilateral: Theorem 5. Level: High School, SAT Prep, College
geometry 
In the figure below ABCD is an equilic quadrilateral.
If AB meet DC in M, equilateral triangles AKC, BJC and BLD are drawn
away from AD, and E and G are the midpoint of the diagonals AC
and BD, prove that K, M, J and L are collinear, J
is the midpoint of KL and EG and KL are parallel lines.


My friend Professor Michael de Villiers
has generalized this result as follows: "If similar triangles
KAC, JBC and LBD are constructed on AC, BC and BD of any
quadrilateral ABCD so that angle AKC = angle AMD, where M is the
intersection of AB and DC extended, then K, M, J and L are
collinear" (allowing for vanishing points collinear on vanishing
line in special cases).
Reference: De Villiers, M. The Role of Proof
in Investigative, Computerbased Geometry: Some personal
reflections. Chapter in Schattschneider, D. & King, J. (1997).
Geometry Turned On! Washington: MAA, pp. 1524.
Some
downloadable Sketchpad 3 sketches from this paper.

