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Published online by Cambridge University Press: 03 November 2016
The theory of the orthocentre is usually considered in conjunction with ordinary triangles only, that is to say, with triangles whose three vertices or sides are distinct. The treatment for the case when two or more sides or vertices coincide is generally overlooked, but it leads to many interesting results and applications. An exposition of some of these is the object of this note.
page 452 note * This result occurs amongst the miscellaneous examples in Russell’s Elementary Treatise on Pure Geometry(Clarendon Press), p. 347, No. 169.
page 453 note * This theorem is stated without proof by R.H. Fowler in The Elementary Differentia Geometry of Curves(Camb. Math. Tracts, (1929) No. 20), p. 31. It is not difficult to show that the radius of the limiting position of the ex-circle is given by 
, where p is the perpendicular from the origin to a tangent and ψ the angle a tangent makes with the x-axis.
page 454 note * This, of course, is well known, v. Russell, loc. cit. p. 347, No. 170.