The as yet unproved quasi-Euler inequality conjecture below (dating back to 2014) is a generalization of the Euler inequality for a bi-centric quadrilateral, that is a quadrilateral which is both inscribed in a circle, as well as circumscribed around another circle, for which R ≥ √2 r and where R is the radius of the circumcircle and r is the radius of the incircle. This interesting inequality can be derived from the formula of Nicolaus Fuss (1755-1826), which precisely relates R to r for a bicentric quadrilateral (Dörrie, 1965). Fuss was a student and friend of Leonhard Euler.
Quasi-Euler Inequality Conjecture
Define for any convex quadrilateral ABCD the quasi-circumradius as R = ½(AO + BO) and the quasi-inradius r = ½(Distance I to AD + Distance I to AB), where O and I are respectively the quasi-circumcentre and quasi-incentre of ABCD. Then R ≥ √2 r, with equality only holding when ABCD is a square.
Definition of quasi-circumcentre
Let K, L, M, and N be the respective intersections of the perpendicular bisectors of the adjacent sides of ABCD. Then the intersection O of the diagonals KM and LN of the quadrilateral KLMN is the quasi-circumcentre of ABCD.
Note as shown in the dynamic sketch below that the quasi-circumcentre O is equi-distant from each pair of opposite vertices of ABCD.
Definition of quasi-incentre
Construct the angle bisectors for each of the four angles of ABCD. 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 diagonals EG and FH of the quadrilateral EFGH is the quasi-incentre of ABCD.
Note as shown in the dynamic sketch below that the quasi-incentre I is equi-distant from each pair of opposite sides of ABCD.
(For more about the quasi-circumcentre and quasi-incentre of a quadrilateral, go to: The quasi-circumcentre and quasi-incentre of a quadrilateral)
Investigate
Use the dynamic sketch below by dragging the vertices to investigate the validity of the inequality conjecture above.
Unproved Conjecture - Quasi-Euler Inequality for Quadrilateral
Challenge
Can you prove the quasi-Euler Inequality stated above? Or can you can disprove it be providing a counter-example?
References
De Villiers, M. (In Press). Another Student Discovery: The Quasi-Circumcentre and Quasi-Incentre of a Quadrilateral. Learning and Teaching Mathematics, No. 39, Dec 2025.
Dörrie, H. (1965). 100 Great Problems of Elementary Mathematics. New York: Dover Publications, pp. 188-192.
Some Related Links
The quasi-circumcentre and quasi-incentre of a quadrilateral
Some other bicentric quadrilateral constructions
Water Supply: Four Towns (introduction to perpendicular bisectors)
Perpendicular Bisectors of Trapezoid (see Investigation 5)
Perpendicular Bisectors of Tangential Quadrilateral
The Perpendicular Bisectors of an Apollonius Quadrilateral
A theorem involving the perpendicular bisectors of a hexagon with opposite sides parallel
Parallelogram Angle Bisectors (Rethinking Proof activity)
Concurrent Angle Bisectors of a Quadrilateral
Side Divider (Wind Screen Wiper) Theorem for a Tangential Quadrilateral
Angle Divider Theorem for a Cyclic Quadrilateral
Conway’s Circle Theorem as special case of Side Divider Theorem
The Equi-inclined Bisectors of a Cyclic Quadrilateral
The Equi-inclined Lines to the Angle Bisectors of a Tangential Quadrilateral
A 1999 British Mathematics Olympiad Problem and its dual
Euler-Nagel line analogy
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
The quasi-Euler line of a quadrilateral and of a hexagon
Diagonal Division Ratios in a Quadrilateral
External Links
Fuss's theorem about a bicentric quadrilateral (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 with JavaSketchpad, 30 November 2014; updated to WebSketchpad, 10 Nov 2025.