CHAPTER 2

Some Properties of Circles

Although the Greeks worked fruitfully, not only in geometry but also in the most varied fields of mathematics, neverthe- less we today have gone beyond them everywhere and cer- tainly also in geometry. F. Klkn

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.

2.1 The power of a point with respect to a circle

We begin our investigations by recalling two of Euclid's theorems: 111.35, 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 X PA' = PB X PB' ), and 111.36, comparing a secant and a tangent drawn from the same point P outside the circle (in Figure 2.1B, PA X PA' = PP ). If we agree to regard a tangent as the limiting form of a secant, we can combine these results as follows: 27 28 PROPERTIES OF CIRCLES

THEOREM2.11. If two lines through a point P meet a circle at points A, A' (possibly coincident) and B, B' (possibly coincident), respec- tively, then PA X PA' = PB X PB' .

Figure 2.1A For a proof we merely have to observe that the similar triangles PA B' and PB A' (with a common angle at P) yield

-=-PA PB PB' PA" In Figure 2.1B, we can equally well use the similar triangles PAT and PTA' to obtain -=-PA PT PT PA" and then say PA X PA' = PP = PB X PB'

Figure 2.1B Let R denote the radius of the circle, and d the distance from P to the center. By taking BB' to be the diameter through P (with B farther from P than B'), we see that, if P is inside the circle (as in Figure 2.1A), APX PA' = BPX PB' = (R+d)(R- d) = R2- 8, and if P is outside (as in Figure 2.1B), EULER'S FORMULA FOR 01

PA X PA' = PB x PB' = (d+ R)(d- R) = 6-R. The equation APX PA' = RZ-dz provides a quick proof of a formula due to Euler: THEOREM2.12. Let 0 and I be the circumcenter and incenter, re- spectively, of a lrMngk with circumradius R and inradius r; kt d be the distance OI. Then (2.12) dL s RZ - 2rR.

Figure 2.1C Figure 2.1C shows the internal bisector of LA extended to meet the circumcircle at L, the midpoint of the arc BC not containing A. LM is the diameter perpendicular to BC. Writing, for convenience, a = +A and @ = $B, we notice that LBML = LBAL = a, and LLBC = LLAC = a. Since the exterior angle of AABI at I is LBIL = a+@ = L LBI, ALBI is isosceles: LI = LB. Thus R-dL = LIXIA = LBXIA sin a = LM -LB/LM Iy = LM- IY I Y/IA sin a = LMXIY = 2121, that is, dL = RZ - 2rR, as we wished to prove. 30 PROPERTIES OF CIRCLES

For any circle of radius R and any point P distant d from the center, we call 6-RZ the paver of P with respect to the circle. It is clearly positive when P is outside, zero when P lies on the circumference, and negative when P is inside. For the first of these cases we have already obtained the alternative expression PA X PAt, where A and A' are any two points on the circle, collinear with P (as in Theorem 2.11). This expression for the power of a point P remains valid for all positions of P if we agree to adopt Newton's idea of directed line segments: a kind of onedimensional vector algebra in which AP = -PA. The product (or quotient) of two directed segments on one line is re- garded as being positive or negative according as the directions agree or disagree. With this convention, the equation 8-RZ = PAXPA' holds universally. If P is inside the circle, 6- R = -(RZ- 6) = -APX PA' = PA X PA'; and if P is on the circumference, either A or A' coincides with P, so that one of the segments has length zero. In fact, after observing that the product PA X PA' has the same value for every secant (or chord) through P, we could have used this value as a definition for the power of P with respect to the circle. The word paver was first used in this sense by Jacob Steiner, whose name has already appeared in Chapter 1.

EXERCISES

1. What is the (algebraically) smallest possible value that the power of a point can have with respect to a circle of given radius R? Which point has this critical power?

2. What is the locus of points of constant power (greater than - Z?) with respect to a given circle?

3. If the power of a point has the positive value P, interpret the length 1 geometrically.

4. If PT and PU are tangents from P to two concentric circles, with T DIRECTED LINE SEGMENTS 31

on the smaller, and if the segment PT meets the larger circle at Q, then PT' - PU = QF.

5. The circumradius of a triangle is at least twice the inradius.

6. Express (in terms of r and R) the power of the incenter with respect to the circumcircle.

7. The notation of directed segments enables us to express Stewart's theorem (Exercise 4 of Section 1.2) in the following symmetrical form [s, p. 152): If P, A, B, C are four Poi& of which the last three are collinear, thm PAZXBC+PPXCA+PCXAB+BCXCAXAB = 0.

8. A line through the centroid G of AABC intersects the sides of the triangle at points X, Y, Z. Using the concept of directed line segments, prove that

9. How far away is the horizon as seen from the top of a mountain one mile high? (Assume the earth to be a sphere of diameter 7920 miles.)

2.2 The radical axis of two circles

The following anecdote was related by E. T. Bell [3, p. 481. Young Princess Elisabeth, exiled from Bohemia, had successfully attacked a problem in elementary geometry by using coordinates. As Bell states it, "The problem is a fine specimen of the sort that are not adapted to the crude brute force of elementary Cartesian geometry." Her teacher was Ren6 Descartes (after whom Cartesian coordinates were namedt). His reaction was that "he would not undertake to carry out her solu- tion . . . in a month." The lesson is clear: a solution that is possible in a certain manner may still not be the best or most economical one. At any rate, here is one theorem for which an analytic proof, without being any more difficult than the usual synthetic proof [6, p. 861, has some interesting re- percussions:

THEOREM2.21. The locus of all points whose powers with respect to two nmmenlric circles are equal is a line perpendicukrr lo the line of centers of the two circles. t There are some who claim that it was Pierre Fermat (1601-1665) who actually invented analytic geometry. Their contention is that he gave the essential idea to Descartes in a letter.