## Nine-point (or Euler) centre & Maltitudes of Cyclic Quadrilateral

1) The perpendiculars from the midpoints of the sides of a cyclic quadrilateral to the opposite sides (called the *maltitudes*) are concurrent in *P*.

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.

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Nine-point (or Euler) centre & Maltitudes of Cyclic Quadrilateral

a) Can you prove the maltitudes result? If stuck, see my paper at *Generalizations involving maltitudes*

b) Can you prove the nine-point (or Euler) centre result? This beautiful result generalizes to any cyclic polygon and a proof is given in Yaglom, I.M. (1968). *Geometric Transformations II*. Washington, DC: The Mathematical Association of America, p. 24.

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Michael de Villiers, 6 April 2010.