1) The perpendiculars from the midpoints of the sides of a cyclic quadrilateral ABCD to the opposite sides (called the maltitudes) are concurrent in P (called the anticentre, nine-point centre or Euler centre by different authors).
2) The four nine-point circles of triangles ABC, BCD, CDA and DAB are also concurrent in the same point P, and is therefore often called the nine-point (or Euler) centre of a cyclic quadrilateral. More-over, the centres of these 4 circles are concyclic, with the centre of the formed circle at P, and all 5 these circles are congruent, with radii half that of the circumcircle of ABCD.
Explore
Use the dynamic sketch below to explore the theorems above by dragging any of the red vertices.
Nine-point centre (anticentre or Euler centre) & Maltitudes of Cyclic Quadrilateral
Challenge
a) Can you prove the maltitudes concurrency result? Also show (prove) that the nine-point centre (anticentre or Euler centre) and the circumcentre are symmetric with respect to the centroid of cyclic quadrilateral ABCD.
b) Can you prove the nine-point (or Euler) centre result involving the concurrency of the four 9-point (Euler) circles & generalize further?
Hint: If stuck, see References below.
Generalization
The beautiful result in b) generalizes to any cyclic polygon, and a proof is given in Yaglom (1968, p. 24).
References
De Villiers, M. (1999). Generalizations involving maltitudes, Int. J. Math. Ed. Sci. Technol, 30(4), pp. 541-548.
Josefsson, M. (2019). Characterizations of Cyclic Quadrilaterals. International Journal of Geometry, Vol. 8, No. 1, pp. 5 - 21.
Mammana, M.F. (2016). Homothetic transformations and geometric loci: properties of triangles and quadrilaterals. International Journal of Mathematical Education in Science and Technology, 47:7, pp. 1103-1119, DOI: 10.1080/0020739X.2016.1164345.
Wells, D. (1991). The Penguin Dictionary of Curious and Interesting Geometry. London, England: Penguin, p. 146.
Yaglom, I.M. (1968). Geometric Transformations II. Washington, DC: The Mathematical Association of America, p. 24.
Some Related Links
The Equi-inclined Bisectors of a Cyclic Quadrilateral
Euler and Nagel lines for Cyclic and Circumscribed Quadrilaterals
Nine Point Conic and Generalization of Euler Line
Cyclic Quadrilateral Alternate Angles Sum (Rethinking Proof activity)
Cyclic Hexagon Alternate Angles Sum Theorem
A generalization of the Cyclic Quadrilateral Alternate Angle Sum theorem
Angle Divider Theorem for a Cyclic Quadrilateral
Conway’s Circle Theorem as special case of Side Divider (Windscreen Wiper) Theorem
Triangle Altitudes Concurrency (Rethinking Proof activity)
Parallelogram Angle Bisectors (Rethinking Proof activity, incl. generalization to any quad)
Napoleon's Theorem (Rethinking Proof activity)
Miquel's Theorem (Rethinking Proof activity)
A variation of Miquel's theorem and its generalization
Cyclic Kepler Quadrilateral Conjectures
Bradley's Theorem for a Circumscribed Quadrilateral
The quasi-circumcentre and quasi-incentre of a quadrilateral
Two British Mathematics Olympiad Concurrency Problems
Similar Parallelograms: A Generalization of a Golden Rectangle property
Japanese theorem for cyclic quadrilaterals
Cyclic Quadrilateral Angle Bisectors Rectangle Result
Cyclic Quadrilateral Midpoints of Arcs Theorem
Brahmagupta's Cyclic Quadrilateral Area Formula
Ptolemy's theorem
External Links
Maltitude (Wolfram MathWorld)
Anticenter (Wolfram MathWorld)
Nine-point circle (Wikipedia)
Quadrilateral Lecture Notes (University of Kentucky, Geometry)
Aimssec Lesson Activities (African Institute for Mathematical Sciences Schools Enrichment Centre)
UCT Mathematics Competition Training Material
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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Michael de Villiers, 6 April 2010; updated with WebSketchpad, 17 October 2021; 14 July 2022; 28 June 2026.