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Equilic
Quadrilateral: Theorem 5. Level: High School, SAT Prep, College
geometry |
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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.
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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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