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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, Computer-based Geometry: Some personal reflections. Chapter in Schattschneider, D. & King, J. (1997). Geometry Turned On! Washington: MAA, pp. 15-24.

Some downloadable Sketchpad 3 sketches from this paper.
 

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