16. It may be shewn, by a geometrical investigation similar to that in Art. 14, that if p1, P2, P, be the semi-diameters of the conic respectively parallel to the sides of the triangle of reference,

2

P12 (wb2 + vc3 — Qu'bc) = p,2 (uc2 + waa — 2v′ca)

Hence, if two conics be similar and similarly situated, the values of the ratios denoted by

wb2+vc2- Qu'bc: uc2+wa-2v'ca: va2 + ub2 — 2w'ab

must be the same for both.

Hence, also, by reasoning similar to that employed in Art. 15, it follows that all conics, similar and similarly situated to each other, intersect in the same two points in the line at infinity.

These points will be real, coincident, or imaginary, according as the conics are hyperbolas, parabolas, or ellipses.

If the conics, in addition to being similar and similarly situated, are also concentric, they will touch one another at the two points where they meet the line at infinity.

17.

To find the radical axis of two similar and similarly situated conics.

By multiplying the equation of one of two given conics by an arbitrary constant, and adding it to the equation of the other given conics, we obtain the general equation of the system of conics passing through their four points of intersection. By suitably determining the arbitrary constant, we may make this equation represent any one of the three pairs of straight lines passing through these four points. In the case, therefore, in which the two conics are similar and similarly situated, it must be possible so to determine the constant that the left-hand member of the equation may break up into two factors, one which equated to zero re

RADICAL AXIS OF TWO CONICS.

89

presents the line at infinity, and the other the radical axis. Hence, if

ua2 + vß2 + wy2 + Qu'ẞy + 2v'ya+2w'aß = 0,

pa2+qB2 +ry2+ 2p′ By + 2q'ya + 2r′aß = 0,

be the equations of two similar and similarly situated conics, it must be possible to determine the arbitrary multiplier k, so that

(u+kp) a2 + (v+ kq) B2 + (w + kr) y3

+2 (u' + kp') By + 2 (v' + kq') ya + 2 (w' + kr') oß

This gives, equating the coefficients of By, ya, aß

(The identity of these three values of k is ensured by the condition of similarity already investigated.)

18. As an example of the application of this formula we may take the following theorem. The nine-point circle of a triangle (that is, the circle which passes through the middle points of its sides) touches each of the four circles which touch the three sides of the triangle.

is the equation of the radical axis of the inscribed and ninepoint circles. The equation of the nine-point circle will then be (see Chap. II. Art. 10),

a2 (s − a)2 a2 + b2 (s — b)2 ß2 + c2 (s — c)2 y2

— 2bc (s—b) (s—c) By — 2ca (s—c) (s — a) ya

— 2ab (s — a) (s — b) aß + (λa + μß +vy) (aa+bB+cy) = 0.

If this represent the nine-point circle, it must be satisfied when a 0 and bẞcy.

Hence

(s — b)2 + (s — c)3 — 2 (s — b) (s — c) + 2

This gives, for the equation of the radical axis,

Now, to ascertain whether this touches the inscribed circle, we have, applying the condition of Chap. II. Art. 9, to investigate the value of

b-c A
Cos2 +

α

{(b − c) (s − a) + (c − a) (s − b) + (a−b) (s—c)},

which is 0. Hence, the radical axis touches the inscribed circle, and therefore the inscribed and nine-point circles touch one another. Similarly, it may be proved that the nine-point circle touches each of the escribed circles.

19. The equation of the nine-point circle may be deduced by substituting the above values of λ, u, v, or (perhaps more neatly) by expressing the fact that the curve

ua2 + vß2 + wy2 + Qu'By + 2v'ya + 2w'aß = 0

passes through the middle points of the sides of the triangle, and combining the equations thus obtained with those investigated in Art. 14. The former gives

Hence, the nine-point circle is represented by the equation a cos A. a2 + b cos B. B2+ccos C. y3 — aẞy — bya— caß = 0.

COR. It hence appears that the nine-point circle passes through the points of intersection of the circumscribed and self-conjugate circles, or has a common radical axis with them.

20. We have investigated, in Art. 10, the equation of the pair of tangents drawn to the conic from a given point (f, g, h). If these two tangents be at right angles to one another, they may be regarded as the limiting form of a rectangular hyperbola, and must therefore satisfy the equation investigated in Art. 13. This, therefore, gives as the locus of the intersection of two tangents at right angles to

one another

Wg2+ Vh2-2U'gh + Uh2+ Wf2-2V 'hf+Vƒ2+ Ug2 - 2 W'fg +2 (U'ƒ3 + Ugh — V'fg - W'hf) cos A

+ 2 (V'g2 + V hf - W'gh — U'fg) cos B

+2(W'h+Wfg-U'hf- V'gh) cos C=0.

This may be shewn (see Art. 15) to represent a circle, as we know ought to be the case.