and, comparing these with the forms of equations of the tangents given in Art. (3), we see that the equation of the circumscribing circle is

7. Having thus discussed the equation of the conic, circumscribing the triangle of reference, we may proceed to investigate that of the conic which touches its three sides. The condition that the conic

ua2 + vß2 + wy2+2u' By +2v'ya+2w′aß = 0,

may touch the line a=0 is, that the left-hand member of the equation obtained by writing a=0 in the above may be a perfect square. This requires that

are necessary conditions that the conic should touch the lines B = 0, y = 0.

We must observe, however, that if the conic touch all three of the sides of the triangle of reference, the three double signs in the above equations must be taken all negatively, or two positively and one negatively. For, if they be taken otherwise, the left-hand member of the equation of the conic will become a perfect square, as may be ascertained by substitution, and the conic will degenerate into a straight line, or rather into two coincident straight lines.

Taking then the double signs all negatively, and writing for convenience, L3, M2, N2, instead of u, v, w, the equation of the conic which touches the three sides of the triangle of reference becomes

L3a2 + M2ß2 + N2y2 – 2MNßy – 2NLyα — 2LMɑß = 0, which is equivalent to

± (La)3± (MB)3 ± (Ny)* = 0.

It may be remarked, that the condition that the point (l, m, n) should lie in the above conic, is the same as the condition that the straight line (l, m, n) should touch the circumscribing conic

LBY + Mya+Naß = 0.

See Art. 5. This we shall return to hereafter.

8. To find the centre of the conic.

Let D, E, F be the points of contact of the sides BC, CA, AB respectively. Join EF, FD, DE, bisect FD, DE in H, I, join BH, CI, and produce them to meet in O. Then O will be the centre of the conic (see p. 32). We have then to find the equations of BH, CI, which, by their intersection, determine 0.

B

H

Fig. 16.

Let f, g, h, be the co-ordinates of D. Then f=0; and 91, h, will be the values of B, y, which satisfy the equations

or

and

1

In like manner it may be proved that, if f2, 92, h2 be the co-ordinates of E,

or

Now, for 1, and therefore for every point in the line CI,

COR. Hence may be obtained the condition that the conic may be a parabola. For the centre of a parabola is infinitely distant, its co-ordinates must therefore satisfy the algebraical relation

This will be observed by reference to Art. 9, to be identical with the condition that the conic should touch the straight line, aa+bB+cy = 0, and thus we are again led to the conclusion noticed in Art. 7, that every parabola touches the line at infinity.

9. To find the condition that the conic should touch a given straight line.

If the straight line (l, m, n) be a tangent to the conic, the values of the ratio : y, obtained by eliminating a between the equation of the conic and the equation

la + mB+ny = 0,

must be equal to one another. For this purpose, it is most convenient to take the equation of the conic in the form

± (La)*± (MB)1± (Ny)* = 0.

CONDITION OF TOUCHING A GIVEN STRAIGHT LINE. 43

Eliminating a, we then get

or

L (mß+ny) +1 {(Mß)1 ± (Ny)3}2 = 0,

(Lm+ M1) ß + (Ln+ N1) y ± 21 (MNßy)3 = 0,

and, if the roots of this, considered as a quadratic in

It hence appears that the condition, that the line (l, m, n) should touch the conic

(La)* ± (MB)*+ (Ny)* = 0,

is identical with the condition that the point (l, m, n) should lie in the conic

a result analogous to that obtained in Art. 13, chap. I.

10.

To find the equations of the four circles which touch the three sides of the triangle of reference.

These may be obtained most readily by the employment of the equations for the determination of the centre, obtained in Art. 8. Thus, let it be required to find the ratios of L, M, N in order that the conic may become the inscribed circle. At the centre of this circle we have, as we know, a=ẞ=y.