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by the relation Σa²(pq) (pr) = 4Δ² (see Geometry: Analytical), which is generally written {ap, bq, cr}² = 4Δ², we obtain {ap, bq, crρ² = 4Δ² {(lp+mq+nr)/(l+m+n)}², the accents being dropped, and p, q, r regarded as current co-ordinates. This equation, which may be more conveniently written {ap, bq, cr}² = (λp + μq + νr)², obviously represents a circle, the centre being λp+μq+νr=0, and radius 2Δ/(λ + μ + ν). If we make λ = μ = ν = 0, ρ is infinite, and we obtain {ap, bq, cr}²=0 as the equation to the circular points.

Systems of Circles.

Centres and Circle of Similitude.—The “centres of similitude” of two circles may be defined as the intersections of the common tangents to the two circles, the direct common tangents giving rise to the "external centre," the transverse tangents to the "internal centre." It may be readily shown that the external and internal centres are the points where the line joining the centres of the two circles is divided externally and internally in the ratio of their radii.

The circle on the line joining the internal and external centres of similitude as diameter is named the "circle of similitude." It may be shown to be the locus of the vertex of the triangle which has for its base the distance between the centres of the circles and the ratio of the remaining sides equal to the ratio of the radii of the two circles.

With a system of three circles it is readily seen that there are six centres of similitude, viz. two for each pair of circles, and it may be shown that these lie three by three on four lines, named the “axes of similitude.” The collinear centres are the three sets of one external and two internal centres, and the three external centres.

Coaxal Circles.—A system of circles is coaxal when the locus of points from which tangents to the circles are equal is a straight line. Consider the case of two circles, and in the first place suppose them to intersect in two real points A and B. Then by Euclid iii. 36 it is seen that the line joining the points A and B is the locus of the intersection of equal tangents, for if P be any point on AB and PC and PD the tangents to the circles, then PA·PB = PC² = PD², and therefore PC = PD. Furthermore it is seen that AB is perpendicular to the line joining the centres, and divides it in the ratio of the squares of the radii. The line AB is termed the “radical axis.” A system coaxal with the two given circles is readily constructed by describing circles through the common points on the radical axis and any third point; the minimum circle of the system is obviously that which has the common chord of intersection for diameter, the maximum is the radical axis—considered as a circle of infinite radius. In the case of two non-intersecting circles it may be shown that the radical axis has the same metrical relations to the line of centres.

EB1911 Circle Fig 5.jpg

There are several methods of constructing the radical axis in this case. One of the simplest is: Let P and P′ (fig. 5) be the points of contact of a common tangent; drop perpendiculars PL, P′L′, from P and P′ to OO′, the line joining the centres, then the radical axis bisects LL′ (at X) and is perpendicular to OO′. To prove this let AB, AB¹ be the tangents from any point on the line AX. Then by Euc. i. 47, AB²=AO²–OB²=AX²+OX²+OP²; and OX²=OD²–DX²=OP²+PD²–DX². Therefore AB²=AX²–DX²+PD². Similarly AB′²=AX²–DX²+DP′². Since PD=PD′, it follows that AB=AB′.

To construct circles coaxal with the two given circles, draw the tangent, say XR, from X, the point where the radical axis intersects the line of centres, to one of the given circles, and with centre X and radius XR describe a circle. Then circles having the intersections of tangents to this circle and the line of centres for centres, and the lengths of the tangents as radii, are members of the coaxal system.

In the case of non-intersecting circles, it is seen that the minimum circles of the coaxal system are a pair of points I and I′, where the orthogonal circle to the system intersects the line of centres; these points are named the "limiting points." In the case of a coaxal system having real points of intersection the limiting points are imaginary. Analytically, the Cartesian equation to a coaxal system can be written in the form x²+y²+2ax±k²=0, where a varies from member to member, while k is a constant. The radical axis is x = 0, and it may be shown that the length of the tangent from a point (0, h) is h² ± k², i.e. it is independent of a, and therefore of any particular member of the system. The circles intersect in real or imaginary points according to the lower or upper sign of k², and the limiting points are real for the upper sign and imaginary for the lower sign. The fundamental properties of coaxal systems may be summarized:—

1. The centres of circles forming a coaxal system are collinear;

2. A coaxal system having real points of intersection has imaginary limiting points;

3. A coaxal system having imaginary points of intersection has real limiting points;

4. Every circle through the limiting points cuts all circles of the system orthogonally;

5. The limiting points are inverse points for every circle of the system.

The theory of centres of similitude and coaxal circles affords elegant demonstrations of the famous problem: To describe a circle to touch three given circles. This problem, also termed the "Apollonian problem," was demonstrated with the aid of conic sections by Apollonius in his book on Contacts or Tangencies; geometrical solutions involving the conic sections were also given by Adrianus Romanus, Vieta, Newton and others. The earliest analytical solution appears to have been given by the princess Elizabeth, a pupil of Descartes and daughter of Frederick V. John Casey, professor of mathematics at the Catholic university of Dublin, has given elementary demonstrations founded on the theory of similitude and coaxal circles which are reproduced in his Sequel to Euclid; an analytical solution by Gergonne is given in Salmon’s Conic Sections. Here we may notice that there are eight circles which solve the problem.

Mensuration of the Circle.

All exact relations pertaining to the mensuration of the circle involve the ratio of the circumference to the diameter. This ratio, invariably denoted by π, is constant for all circles, but it does not admit of exact arithmetical expression, being of the nature of an incommensurable number. Very early in the history of geometry it was known that the circumference and area of a circle of radius r could be expressed in the forms 2πr and πr². The exact geometrical evaluation of the second quantity, viz. πr², which, in reality, is equivalent to determining a square equal in area to a circle, engaged the attention of mathematicians for many centuries. The history of these attempts, together with modern contributions to our knowledge of the value and nature of the number π, is given below (Squaring of the Circle).

The following table gives the values of this constant and several expressions involving it:—

  Number.  Logarithm.    Number.  Logarithm. 
π 3.1415927 0.4971499  π² 9.8696044 0.9942997
2π 6.2831858 0.7981799
4π 12.5663706 1.0992099 1/6π² 0.0168869 ̅2.2275490
1/2π 1.5707963 0.1961199
1/3π 1.0471976 0.0200286  √π 1.7724539 0.2485750
1/4π 0.7853982 ̅1.8950899
1/6π 0.5235988 ̅1.7189986 3π 1.4645919 0.1657166
1/8π 0.3926991 ̅1.5940599
1/12π 0.2617994 ̅1.4179686 1/π 0.5641896 ̅1.7514251
4/3π 4.1887902 0.6220886
π/180 0.0174533 ̅2.2418774 2/π 1.1283792 0.0524551
1/π 0.3183099 ̅1.5028501 1/2√π 0.2820948 ̅1.4503951
4/π 1.2732395 0.1049101 3√(6/π) 0.2820948 1.4503951
1/4π 0.0795775 ̅2.9097901 3√(3/4π) 0.6203505 ̅1.7926371
180/π 57.2957795 1.7581226 loge π 1.1447299 0.0587030

Useful fractional approximations are 22/7 and 355/113.

A synopsis of the leading formula connected with the circle will now be given.

1. Circle.—Data: radius = a.  Circumference = 2πa.  Area = πa².

2. Arc and Sector.—Data: radius = a; θ = circular measure of angle subtended at centre by arc; c = chord of arc; c2 = chord of semi-arc; c4 = chord of quarter-arc.