The quasi-circumcentre and quasi-incentre of a quadrilateral

The first result below about the 'quasi-circumcentre' was experimentally discovered and proved from a problem posed in De Villiers (1999) 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 - an online version of the original problem is available at: Water Supply: Four Towns (click on the 'Link to Water Supply: Finding Minimum' button).
It followed from the classroom discussion of a proposed solution for this problem by an undergraduate student, Renate Lebleu Davis, at Kennesaw State University during 2006 - see De Villiers (in press). (This result was also later 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 (up to Grade 9) in Durban in 2009).

Renate's Theorem about the quasi-circumcentre of a quadrilateral
Given a quadrilateral ABCD as shown below. Let K, L, M, and N be the respective circumcentres of triangles ABD, ABC, BCD, and CDA. (Alternatively, let K, L, M, and N be the respective intersections of the perpendicular bisectors of the adjacent sides of ABCD. For example, K is the intersection of the perpendicular bisectors of sides AD and AB, etc.)
1) What do you notice about the distances of the intersection O of the diagonals KM and LN to the vertices of ABCD?
2) Do the distance relationships you notice remain valid if the quadrilateral becomes concave or crossed?
3) Investigate by dragging the vertices of ABCD and formulate a conjecture.

Quasi-circumcentre & Quasi-incentre of quadrilateral

Conjecture 1
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.

Challenge 1
4) Can you explain why (prove that) Conjecture 1 is true?

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The side-angle duality
Given the interesting side-angle duality in Euclidean plane geometry observed and explored in De Villiers (1994, 1996), and specifically the duality between perpendicular bisectors and angle bisectors, it was natural to next explore the following analogous result.

Theorem 2: The quasi-incentre of a quadrilateral
Given a quadrilateral ABCD as in the second figure above (navigate to it by clicking on the 'Link to quasi-incentre' 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.
5) What do you notice about the distances from I, the intersection of EG and FH, to the sides of ABCD?
6) Do the distance relationships you notice remain valid if the quadrilateral becomes concave or crossed?
7) Investigate by dragging the vertices of ABCD and formulate a conjecture.

Conjecture 2
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.

Challenge 2
8) Can you explain why (prove that) Conjecture 2 is true?

Further Property
Also note that the quadrilateral EFGH in Theorem 2 is a cyclic quadrilateral - see for example, the Parallelogram Angle Bisectors activity.

References
De Villiers, M. (1994, 1996, 2009). Some Adventures in Euclidean Geometry (free to download). Dynamic Mathematics Learning, Lulu Press.
De Villiers, M. (1996). An Interesting Duality in Geometry. AMESA Proceedings, Peninsula Technikon, 1-5 July 1996, pp. 345-350.
De Villiers, M. (1999, 2003, 2012). Rethinking Proof with Geometer's Sketchpad (free to download). Key Curriculum Press.
De Villiers, M. (2014). Quasi-circumcenters and a Generalization of the Quasi-Euler Line to a Hexagon. Forum Geom., 14, 233–236.
Myakishev, A. (2006). On Two Remarkable Lines Related to a Quadrilateral. Forum Geom., 6, 289–295.

Proofs
Both results above about the quasi-circumcentre and quasi-incentre can easily be proved using the respective properties of perpendicular bisectors and angles bisectors, e.g. see proofs.

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Further generalization & application
9) Can you generalize the above theorems further or apply them in other contexts? Below are some generalizations.

Theorem 3: Perpendicular bisectors hexagon concurrency
(The following result was experimentally discovered around 2008/2009 using dynamic geometry).
10) Click on the 'Link to quasi-circumcentre hexagon concurrency' button to navigate to a new sketch.
If the quasi-circumcenters P, Q, R, S, T, and U, respectively of the quadrilaterals ABCD, BCDE, CDEF, DEFA, EFAB, and FABC subdividing an arbitrary hexagon ABCDEF are constructed, then the lines connecting opposite vertices of the hexagon formed by these quasi-circumcenters are concurrent.
Alternative formulation: Let P be the intersection of the perpendicular bisectors of AC and BD, Q the intersection of perpendicular bisectors BD and CE, R the intersection of perpendicular bisectors CE and DF, S the intersection of perpendicular bisectors DF and EA, T the intersection of perpendicular bisectors EA and FB and U the intersection of perpendicular bisectors FB and AC, then the lines connecting opposite vertices of the hexagon formed by these points are concurrent.
11) Does the concurrency remain valid if the quadrilateral becomes concave or crossed? Investigate by dragging the vertices of ABCD

Challenge 3
12) Can you explain why (prove that) Theorem 3 is true?
Note: A proof of this result, using the converse of Pappus, is given in De Villiers (2014).

Theorem 4: Angle bisectors hexagon concurrency
(Given the side-angle duality referred to earlier, it seems natural & intuitive to anticipate a similar result involving angle bisectors of a hexagon constructed in analogous way to those of the perpendicular bisectors. Here it is the 2nd formulation of Theorem 3 that comes in handy).
13) Click on the 'Link to quasi-incentre hexagon concurrency' button to navigate to a new sketch.
Let P be the intersection of the angle bisectors of angles EAC and FDB, Q the intersection of angle bisectors of angles FDB and ACE, R the intersection of angle bisectors ACE and BDF, S the intersection of perpendicular bisectors BDF and CEA, T the intersection of perpendicular bisectors CEA and DFB and U the intersection of perpendicular bisectors DFB and EAC, then the lines connecting opposite vertices of the hexagon formed by these points are concurrent.
14) Does the concurrency remain valid if the quadrilateral becomes concave or crossed? Investigate by dragging the vertices of ABCD

Challenge 4
15) Can you explain why (prove that) Theorem 4 is true?
Note: A proof of this result, using the converse of Pappus, can be given similar to that in De Villiers (2014) for Theorem 3.

Corollaries
Of some further interest too, is that both Theorems 3 and 4 have interesting corollaries in that the formed hexagons are tangential (circumscribed) to a conic. This follows directly from the concurrency of the main diagonals and the converse of Brianchon's theorem.

Note
Note as shown by the measurements in both dynamic sketches for Theorems 3 and 4, the points of concurrency are not necessarily equi-distant from a pair of opposite vertices or opposite sides.

The quasi-Euler line of a quadrilateral and a hexagon
The concept of the Euler line was ingeniously extended to a quadrilateral by Myakishev (2006) to show that the quasi-circumcentre is collinear with the so-called quasi-orthocentre and centre of gravity. This idea was extended further to a hexagon by De Villiers (2014) - see for example the dynamic version of the The quasi-Euler line of a quadrilateral and of a hexagon.

External Links
Pappus's hexagon theorem (Wikipedia)
Brianchon's theorem (Wikipedia)
SA Mathematics Olympiad
Questions and worked solutions for past South African Mathematics Olympiad papers can be found at this link.
(Note, however, that prospective users will need to register and log in to be able to view past papers and solutions.)

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By Michael de Villiers. Created, 23 November 2014; updated 28 July 2020; updated to WebSketchpad, 26 June 2025; updated 30 June 2025; 8 August 2025..