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Triangles With Common Circumcentre and Orthocentre

Published online by Cambridge University Press:  03 November 2016

D.G. Taylor
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Extract

1. The object of this paper may be best explained by reference to the figure. ABC is a triangle with circumcentre O and orthocentre P. What other triangles share these points with ABC?

There is clearly a doubly infinite family of such triangles. If we confine attention for the moment to those possessing the same circumradius R, we shall have a singly infinite family, which we may call the family R. These also possess in common, in addition to circumcentre and orthocentre,

(i) nine-points centre U (the middle point of OP) and nine-points circle, radius ½R ;

(ii) centroid G, where (OGUP) is harmonic.

Type
Research Article
Information
The Mathematical Gazette , Volume 39 , Issue 328 , May 1955 , pp. 106 - 108
Copyright
Copyright © The Mathematical Association 1955

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References

* See, e.g., Lock and Child, A new trigonometry for schools and colleges, (Macmillan, 1911) p. 3

* Hilton, Plane algebraic curves, (Oxford, 1920), p. 319.