Let ,
, , and be four points on a circle ,
and ,
, , the orthocenters of triangles , 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 with regard to triangle is the same as that of
with regard to the triangle .

See also Angle Bisector ,

Concyclic ,

Cyclic Polygon ,

Cyclic
Quadrilateral ,

Orthocentric System
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References 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. Referenced on Wolfram|Alpha Cyclic Quadrangle
Cite this as:
Weisstein, Eric W. "Cyclic Quadrangle."
From MathWorld --A Wolfram Web Resource. https://mathworld.wolfram.com/CyclicQuadrangle.html

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