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A point P in the plane of triangle ABC is said to be an isoperimetric point if PA + PB + AB = PB + PC + BC = PC + PA + CA, and is said to be a point of equal detour if PA + PB − AB = PB + PC − BC = PC + PA − CA. Incorrect conditions for the existence and uniqueness of such points were given by G. R. Veldkamp in Amer. Math. Monthly 92 (1985) 546-558. In this paper, we use a much simpler approach that yields correct versions of these conditions and that exhibits the relations of these points to the centers of the Soddy circles.

The exploration of curves that seem to appear in drawings made of straight lines is fascinating and interesting to learners from the very young to the quite mature. Consistent with the Curriculum and Evaluation Standards for School Mathematics, problem solving, reasoning, communication, and connections are integrated into mathematics learning activities that permit students to know mathematics by doing mathematics. Examples in this article show how students may create drawings of sets of lines whose envelopes are parabolas, cardioids, epicycloids, or hypocycloids.

(1963). An Analogue of the Brocard Points. The American Mathematical Monthly: Vol. 70, No. 5, pp. 495-501.

All elements mentioned herein are in a group G. A well-known definition states that x and y are conjugate if there exists an element a such that y = α–1xα. Conjugacy is an equivalence relation in G. In the present paper this will be called 1-conjugacy.

(1960). On the Brocard Points of a Triangle. The American Mathematical Monthly: Vol. 67, No. 6, pp. 520-525.