The circle has been held in highest esteem through the ages. Its perfect form has affected philosophers and astronomers alike. Until Kepler derived his laws, the thought that planets might move in anything but circular paths was unthinkable. Nowadays, the words “square”, “line”, and the like sometimes have derogatory connotations, but the circle—never. Cleared of superstitious nonsense and pseudo-science, it still stands out, as estimable as ever.

Limitations of space make it impossible for us to present more than a few of the most interesting properties developed since Euclid of the circle and its relation to triangles and other polygons.

The power of a point with respect to a circle

We begin our investigations by recalling two of Euclid's theorems: III.35, about the product of the parts into which two chords of a circle divide each other (that is, in the notation of Figure 2.1A, PA × PA′ = PB × PB′), and III.36, comparing a. secant and a tangent drawn from the same point P outside the circle (in Figure 2.1B, PA × PA′ = PT2).