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Quasi-circumcentre: Given a quadrilateral ABCD as shown below. Let K, L, M, and N be the respective circumcentres of triangles ABD, ADC, BCD and ABC, then the intersection O of KM and LN is equidistant from opposite vertices A and C, as well as equidistant from opposite vertices B and D. The point O is called the quasi-circumcentre of ABCD.
Quasi-circumcentre of quadrilateral
Renate's Theorem: The above result about the quasi-circumcentre was experimentally discovered and proved from a problem posed in  to find the “best” place to build a water reservoir for four villages of more or less equal size, if the four villages are not concyclic. It followed from the classroom discussion of a proposed solution by an undergraduate student, Renate Lebleu Davis, at Kennesaw State University during 2006.
This result was used in the Kennesaw State Mathematics Competition for High School students in 2007, as well as in the World InterCity Mathematics Competition for Junior High School students in 2009.
Reference:  M. de Villiers, Rethinking Proof with Sketchpad, Emeryville: Key Curriculum Press, 1999/2003.
Quasi-incentre: Given a quadrilateral ABCD as shown below. Construct the angle bisectors for each of the four angles. Label E the intersection of the angle bisectors of angles A and B, label F the intersection of the angle bisectors of angles B and C, label G the intersection of the angle bisectors of angles C and D, and label H the intersection of the angle bisectors of angles D and A. Then I the intersection of EG and FH is equidistant from opposite sides AD and BC, as well as equidistant from opposite sides AB and CD. The point I is called the quasi-incentre of ABCD
Quasi-incentre of quadrilateral
Comment: Note that both above results remain valid if ABCD is concave or crossed - check by dragging!
Both results above can easily be proved using the respective properties of perpendicular bisectors and angles bisectors, e.g. see proofs.
Further generalization & application: The quasi-circumcentre can be extended to a quadrilateral and a hexagon where it is collinear with the so-called quasi-orthocentre of a quadrilateral and centre of gravity, lying on the so-called quasi-Euler line of a quadrilateral and a hexagon.
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By Michael de Villiers. Created, 23 November 2014.