and CC' is the polar of the point P (15, Cor. and 18). In like manner it may be shown that the polar of any other point in DP passes through the pole C.

XXIV.

Theorem. Conversely, if any number of lines pass through one point, the locus of their poles is the polar of the point.

Let CC' be any line passing through the point C, whose polar is DP. Draw OC'P CC'. Then (as in 23) we find OC′ ×OP= the square of the radius: hence P is the pole of CC'; and the pole of any line passing through C is in its polar, PD.

XXV.

Since two points determine the position of a straight line, it follows from the preceding articles, that the line joining any two points is the polar of the intersection of the polars of those points, and, conversely, the intersection of two lines is the pole of the line joining the poles of those lines.

COR. 1. If two sides of a triangle are the polars of the opposite vertices with respect to a circle, the third side is the polar of the opposite vertex with respect to the same circle.

For if A is the pole of BC, and B is the pole of AC, the intersection, C, is the pole of AB.

DEF. Any triangle, ABC, whose vertices are the poles of the opposite sides, is said to be self-reciprocal with respect to the circle.

COR. 2. It is evident from the definition of poles and polars, that the altitudes of a self-reciprocal triangle pass through the centre of the circle.

A

D

B

RECIPROCAL POLARS.

XXVI.

Theorem. If two polygons are so related to each other with respect to a circle, that every vertex of one is the pole of a corresponding side of the other, or conversely, then each vertex of the latter is the pole of a corresponding side of the former.

For if A, B, C, D, &c., the vertices of one polygon, are the poles of a, b, c, d, &c., of the other, respectively, then AB is the polar of the intersection of a and b, &c. (25).

DEF. Two polygons thus related to each other with respect to a circle are called reciprocal polars; and the circle is called the auxiliary circle.

COR. 1. Since the pole of a tangent is its point of contact, it follows that the reciprocal polar of an inscribed polygon is the circumscribed polygon formed by drawing tangents at the vertices.

COR. 2. It is evident, from the definition of poles and polars, that the lines drawn from the vertices, A, B, C, D, &c., of either one of two reciprocal polars perpendicular to a, b, c, d, &c., the sides of the other, will pass through the centre of the auxiliary circle. (Construct the figure.)

COR. 3. The theorem is true when the polygons have an infinite number of sides; that is, when they are curves. The infinitesimal sides prolonged are tangents: hence, if any two curves are so related to each other with respect to a circle, that every point of one is the pole of a corresponding tangent to the other, and conversely, then every point of the latter is the pole of a corresponding tangent of the former.

EXERCISE 1. Two concentric circles are reciprocal polars with respect to a third concentric circle, when the radius of the latter is a geometrical mean between the radii of the former.

EXERCISE 2. If A, B, C, D, &c., are the vertices of a polygon, and a, b, c, d, &c., the corresponding sides of its recip

rocal polygon with respect to a circle, find the pole of the diagonal AD.

XXVII.

The relation between two reciprocal polars is such that every line of the one has a corresponding point in the other, and conversely: hence it follows, that, from any theorem in relation to the [] of one figure, there follows reciprocally a corresponding theorem in relation to the [] of the other.

lines

points

lines

By constructing an inscribed hexagon and its reciprocal polar (26, Cor. 1), it will be seen that Brianchon's theorem (13) and corollaries follow reciprocally from Pascal's theorem and corollaries (12), and conversely. For the points P, Q, and R (see fig. in 12) are the poles of the three diagonals joining the opposite vertices of the reciprocal polar of ABCDEF. Hence these diagonals all pass through the pole of PQR (23).

XXVIII.

Any two reciprocal polars possess the following properties:

1. Every line of one is perpendicular to the line joining the corresponding point of the other, and the centre of the circle; and conversely.

2. The angle formed by two lines is equal to (supplement of) the angle contained by the two lines drawn from their poles to the centre of the auxiliary circle (I., 35, Cor. 5); and conversely.

3. The product of the distances of any line of one and the corresponding point of the other from the centre of the auxiliary circle is constant; and conversely (15).

4. If two points of one are equidistant from the centre of the auxiliary circle, the corresponding lines of the other are also equidistant from the centre of the circle.

5. If three or more points of one are in a straight line, the

corresponding lines of the other pass through a common

point.

From

and 2 there follows,

6. If four points are in a straight line, their polars with respect to a circle form a pencil which has the same anharmonic ratio as the points.

XXIX.

When a circle is concentric with the auxiliary circle, its reciprocal polar is also a circle. When it is not concentric with the auxiliary circle, the reciprocal polar is a figure of a more general form: hence we may infer from simple properties of the circle all the reciprocal properties of the more complicated figures into which the circle may be transformed by the process of reciprocation.

XXX.

SALMON'S THEOREM.

Theorem. The distances of any two points from the centre

AO: BO

OA X OC=OB X OD=square of radius,
OD: OC=Oa+AD: 0b+BC;
Oa: Ob (III., 22):

Then

whence

but

AO: BO

=

Construct the figures when one or both the points A and B are without the auxiliary circle.

RADICAL AXIS OF CIRCLES.

XXXI.

The power of a point with respect to a circle = product of the segments of the secants drawn through the point = the square of the distance of the point from the centre minus the square of the radius = the square of the tangent drawn from the point to the circle (III., 37, and corollaries).

DEF. If P be a point in the line AB joining the centres of two circles, whose radii are R and r, such that AP-BP2 = R2-r2, the line PD AB is called the radical axis of the circles. (See next fig.)

COR. The powers of the point P with respect to the two circles are equal. For AP2-R2 — BP2— „2.

From the definition of the radical axis, it is evident

1. If the circles are equal, and not concentric, it bisects the line joining the centres.

2. If the circles are concentric, and not equal, it is at an infinite distance.

3. If the circles are concentric and equal, it is indeterminate.

4. If the circles intersect, it passes through the two points of intersection (II., 14).

5. If the circles touch each other, it is their common tangent.

XXXII.

Theorem. The locus of all the points whose powers with respect to two circles are equal is the radical axis of the circles.

HYPOTH. R and r are the radii of two circles, whose centres are A and B. D is any point whose distances DA and DB are such that DA2 — R2 — DB2 — p2.