The quasi-circumcentre and quasi-incentre of a quadrilateral

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1) Given a quadrilateral ABCD as shown below. Let K, L, M, and N be the respective circumcentres of triangles ABD, ADC, BCD and ABC, what do you notice about the intersection O of KM and LN?
Challenge 1: Can you explain why (prove) your observation is true?

Quasi-circumcentre & Quasi-incentre of quadrilateral

You should have noticed that 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. We shall call point O the quasi-circumcentre of ABCD.

Renate's Theorem: The first result above about the quasi-circumcentre was experimentally discovered and proved from a problem posed in [1] 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 Durban in 2009.

Reference: [1] M. de Villiers, Rethinking Proof with Sketchpad, Emeryville: Key Curriculum Press, 1999/2003.

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2) Given a quadrilateral ABCD as in the second figure above (navigate to it using the LINK button). 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. What do you notice about I be the intersection of EG and FH?
Challenge 2: Can you explain why (prove) your observation is true?

You should have noticed that 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. We shall call the point I the quasi-incentre of ABCD.

Comment: Note that both above results remain valid if ABCD is concave or crossed - check by dragging!

Further generalization & application: The concept of quasi-circumcentre can be extended to a quadrilateral and a hexagon, respectively, where it is collinear with the so-called quasi-orthocentre and centre of gravity, lying respectively on the so-called quasi-Euler line of a quadrilateral and of a hexagon.