line ẞ sin A+ a sin B meets the circle in two coincident points, it is (Art. 83) a tangent at the point aß.

We saw (Art. 63) that a sin A + B sin B is the equation of a parallel to the base (7) drawn through the vertex aß. Hence, by Art. 57, the tangent a sin B + ẞ sin A makes the same angle with one side that the base makes with the other (Euclid, 111. 32). From the forms of the equations of the three tangents,

it appears, that the three points in which they intersect each the opposite side are in one right line, whose equation is

It will be found that the equations of the lines joining the vertices of the inscribed triangle to those of the circumscribed are

and these meet in a point (Art. 36).

108. We shall next show how to obtain the equation of the circle inscribed in the triangle a, ß, y. The equation

l2a2 + m2ß2 + n2y2 – 2mnẞy ·
2mnßy - 2nlya - 2lmaß = 0,

represents a curve of the second degree, touching each of the lines a, ß, y; for if we seek the point where any side (y) cuts the figure, making y = 0, we obtain the perfect square,

l'a2 + m2ß2 - 2lmaß = 0;

the roots of this equation being equal, we infer that the two points coincide in which y cuts the figure, and therefore (Art. 83) that y is a tangent.

In the same manner it can be proved that the sides a and B touch the curve represented by the preceding equation.

This equation may also be written in a convenient form,
l*a* + m3ß* + n1y* = 0;

for if we clear this equation of radicals, we shall find it to be identical with that just written.

For the simplest method of obtaining the particular values of l, m, n, for which the preceding equation represents a circle, I am indebted to Dr. Hart, who derives the equation of the inscribed circle from that of the circumscribed, as follows: Join the points of contact of the circle inscribed in a triangle; let the equations of the sides of the triangle so formed be a' = 0, ß' = 0, y' = 0, and its angles A', B', C'; then (Art. 105) the equation of the circle must be

B'y' sin A' + y'a' sin B' + a'ẞ' sin C'

= 0.

Now we have proved (Art. 104) that for any point on the circle

a2 = By; B'2 = ya ;

12

=

αβ,

and it is easy to see that

A' = 90° - A ; B' = 90° - B; C′ = 90° - C. Substituting these values, the equation of the circle becomes

a cos A+ B+ cos B + y cos C = 0.

The general equation will, therefore, represent a circle if l, m, n, be proportional to

cos2 A, cos B, cos2 C.

It can be proved, in like manner, that the equation of the circle touching the side a, and the sides b and c produced, is

acos A+ B+ sin 3B + y3 sin C = 0.

109. Since the general equation given in the last article may be written in the form

ny (ny - 2la - 2mß) + (la − mẞ)2 = 0,

it follows that the line (la - mẞ), which obviously passes through the point aß, passes also through the point where y meets the curve. The three lines, then, which join the points of contact of the sides with the opposite angles of the circumscribing triangle ny - la = 0,

are

la - mẞ = 0, mß - ny = 0,

and these obviously meet in a point.

The very same proof which showed that y touches the curve shows also that ny 2la - 2mẞ touches the curve, for when this

quantity is put = 0, we have the perfect square (la - mẞ)2 = 0; hence this line meets the curve in two coincident points, that is, touches the curve, and la – mß passes through the point of contact. Hence, if the vertices of the triangle be joined to the points of contact of opposite sides, and at the points where the joining lines meet the circle again, tangents be drawn, their equations are 2la + 2mẞ – ny = 0, 2mß + 2ny - la +2ny - la = 0, 2ny + 2la - mẞ = 0. mß Hence we infer that the three points, where each of these tangents meets the opposite side, lie in one right line, la + mẞ + ny = 0,

for this line passes through the intersection of the first line with y, of the second with a, and of the third with B.

CHAPTER IX.

PROPERTIES OF A SYSTEM OF TWO OR MORE CIRCLES.

110. To find the equation of the chord of intersection of two circles.

If S = 0, S'= 0, be the equations of two circles, then any equation of the form SkS' = 0 will be the equation of a figure passing through their points of intersection (Art. 36).

Let us write down the equations

=

and it is evident that the equation SkS' 0 will in general represent a circle, since the coefficient of ry = 0, and that of x2= that of y2. There is one case, however, where it will represent a right line, namely, when k = 1. The terms of the second degree then vanish, and the equation becomes

S - S' = 2(a' − a) x + 2 (b′ − b)y + r'2 = p2 + a2 − a'2 + b2 − b'2 = 0.

This is, therefore, the equation of the right line passing through the points of intersection of the two circles.

111. The points of intersection of the two circles are found by seeking, as in Art. 80, the points in which the line SS' meets either of the given circles. These points will be real, coincident, or imaginary, according to the nature of the roots of the resulting equation; but it is remarkable that, whether the circles meet in real or imaginary points, the equation of the chord of intersection, S-S' = 0, always represents a real line, having important geometrical properties in relation to the two circles. This is in conformity with our assertion (Art. 81), that the line joining two points may preserve its existence and its properties when those points have become imaginary.

In order to avoid the harshness of calling the line S - S' = 0 the chord of intersection in the case where the circles do not geometrically appear to intersect, it has been called* the radical axis of the two circles.

112. One of the most remarkable properties of this line is found by examining the geometric meaning of the equation S-S'0. We saw (Art. 88) that if the co-ordinates of any point ay be substituted in S, it represents the square of the tangent drawn to the circle S, from the point xy. square of the tangent drawn to the circle S', SS' = 0 asserts, that if from any point on the radical axis tangents be drawn to the two circles, these tangents will be equal.

So also S' is the and the equation

The line (S S') possesses this property whether the circles meet in real points or not. When the circles do not meet in real points, the position of the radical axis is determined geometrically by cutting the line joining their centres, so that the difference of the squares of the parts may = the difference of the squares of the radii, and erecting a perpendicular at this point; as is evident, since the tangents from this point must be equal to each other.

If it were required to find the locus of a point whence tangents to two circles have a given ratio, it appears, from Art. 88, that the equation of the locus will be

S- k2S' = 0,

* By M. Gaultier of Tours (Journal de l'École Polytechnique, Cahier xvi. ; 1813).

which (Art. 110) represents a circle passing through the real or imaginary points of intersection of S and S'. When the circles S and S' do not intersect in real points, we may express the relation which they bear to the circle SkS' by saying that the three circles have a common radical axis.

113. From the form of the equation of the radical axis of two circles, we at once derive the following theorem :

Given any three circles, if we take the radical axis of each pair of circles, these three lines will meet in a point, and this point is called the radical centre of the three circles.

For the equations of the three radical axes are

S-S' 0, S-S" 0, S" - S = 0,

=

which, by Art. 37, meet in a point.

From this theorem we immediately derive the following: If several circles pass through two fixed points, their chord of intersection with a fixed circle will pass through a fixed point.

For, imagine one circle through the two given points to be fixed, then its chord of intersection with the given circle will be fixed; and its chord of intersection with any variable circle drawn through the given points will plainly be the fixed line joining the two given points. These two lines determine, by their intersection, a fixed point through which the chord of intersection of the variable circle with the first given circle must pass.

Ex. 1. Find the radical axis of

x2 + y2 — 4x − 5y + 7 = 0; x2 + y2 + 6x + 8y − 9 = 0.

Ex. 2. Find the radical centre of

Ans. 10x+13y=16.

(x-1)2+(y-2)2 = 7; (x − 3)2 + y2 = 5'; (x + 4)2 + (y + 1)2 = 9.

114. A system of circles having a common radical axis possesses many remarkable properties which are more easily investigated by taking the radical axis for the axis of y, and the line joining the centres for the axis of r. Then the equation of any circle will be x2 + y2 2kx + S2 = 0,

where is the same for all the circles of the system, and the

P