Cyclic Quadrangle

Let A_1, A_2, A_3, and A_4 be four points on a circle, and H_1, H_2, H_3, H_4 the orthocenters of triangles DeltaA_2A_3A_4, etc. If, from the eight points, four with different subscripts are chosen such that three are from one set and the fourth from the other, these points form an orthocentric system. There are eight such systems, which are analogous to the six sets of orthocentric systems obtained using the feet of the angle bisectors, orthocenter, and polygon vertices of a generic triangle.

On the other hand, if all the points are chosen from one set, or two from each set, with all different subscripts, the four points lie on a circle. There are four pairs of such circles, and eight points lie by fours on eight equal circles.

The Simson line of A_4 with regard to triangle DeltaA_1A_2A_3 is the same as that of H_4 with regard to the triangle DeltaH_1A_2A_3.

See also

Angle Bisector, Concyclic, Cyclic Polygon, Cyclic Quadrilateral, Orthocentric System

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Coxeter, H. S. M. and Greitzer, S. L. "Cyclic Quadrangles; Brahmagupta's Formula." §3.2 in Geometry Revisited. Washington, DC: Math. Assoc. Amer., pp. 56-60, 1967.Johnson, R. A. Modern Geometry: An Elementary Treatise on the Geometry of the Triangle and the Circle. Boston, MA: Houghton Mifflin, pp. 251-253, 1929.

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Cyclic Quadrangle

Cite this as:

Weisstein, Eric W. "Cyclic Quadrangle." From MathWorld--A Wolfram Web Resource.

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