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Published online by Cambridge University Press: 20 January 2009
§ 1. The famous theorem of the pedal line of a triangle in ordinary geometry can be stated as follows:—“Given a triangle ABC and a point P such that the feet of the perpendiculars X, Y, Z, dropped from P on the sides of the triangle, are collinear, then the locus of P is the circumcircle.” In noneuclidean geometry this locus is not a circle or even a curve of the second degree, but a cubic; and in both cases the envelope of the line XYZ is a curve of the third class. The explanation of the inconsistency in ordinary geometry is that the complete locus consists of the circumcircle together with the straight line at infinity.
* See the author's paper, “The pedal line of the triangle in non-euclidean geometry.” Proc. Intern. Cong. Math. 5 (Cambridge, 1912), ii., 93–101.Google Scholar
* An alternative equation resulting from this equation is
This corresponds to the condition that 
* The point O′ lies on both cubics, but is not a self-corresponding point. The p-correspondent of O′ is O 1 and the P-correspondent of O′ is T′, the tangential of O′ (see § 16).
* The coordinates of P′, the opposite of P ≡ (x, y, z), are found to be
the values of y′ and z′ being written down by cyclic permutation of all the letters. If (p, q, r) is any point, the join of opposites will pass through this fixed point if
i.e. if (x, y, z) lies on a certain cubic. Similarly, the join of a pair of isogonal or isotomio conjugates will pass through a fixed point only if the points lie on a certain cubic.
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