Similarly, the equations of the tangents at B and C are

4. To determine the position of the centre of the conic. Through the angular points A, B, C of the triangle of reference draw the tangents EAF, FBD, DCE.

Fig. 14.

Bisect

AC, AB respectively in H, I, join EH, FI, and produce them to intersect in O. Then, since every straight line drawn through the intersection of two tangents so as to bisect their chord of contact passes also through the centre, O will be the centre of the conic.

COR. We may hence deduce the relation which must hold between X, u, v, in order that the conic may be a parabola. For, since the centre of a parabola is at an infinite distance, its co-ordinates will satisfy the equation

aa+bB+cy = 0.

We hence obtain the following equation:

λ3a2 + μ3b2 + v2c2 - 2μvbc - 2vλca — 2λμab = 0,

which is equivalent to

± (λa)* ± (μb)1 ± (vc)* = 0,

as the necessary and sufficient condition that the conic should be a parabola.

5. To determine the condition that a given straight line may touch the conic.

If the conic be touched by the straight line (l, m, n), the two values of the ratio By, obtained by eliminating a between the equations

λβη + μιγα + ναβ = 0,

la+mB+ny = 0,

must be coincident. The equation which determines these is - λίβη + (μη + νβ) (mβ + γ) = 0,

and the condition that the two values of ß: y be equal, is 4μn. vm - (μm + vn - xl)2 = 0,

or X2l2 + μ3m2 + v3n3 — 2μv. mn — 2vλ . nl — 2λμ. lm = 0,

which is equivalent to

± (Xl)* ± (μm)* ± (vn)3 = 0.

If this be compared with the condition investigated in Art. (4) that the conic may be a parabola, it will be observed that the parabola satisfies the analytical condition of touching the straight line aa+b+cy = 0. This is generally expressed by saying that every parabola touches the line at infinity.

6. To investigate the equation of the circle, circumscribing the triangle of reference.

This may be deduced from the consideration that the co-ordinates of the centre of the circumscribing circle are respectively proportional to cos A, cos B, cos C (see p. 4). Or

it may be independently investigated as follows. Draw EAF, FAD, DAE (fig. 2), tangents to the circle, then the angle EAC is equal to ABC, and FAB to ACB (Euc. III. 32). Hence the equation of the tangent EAF must be

are

F

B

+ =

0.

E

Similarly the equations of the other tangents FBD, DCE

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 + Qu' By + 2v'ya + 2w'aß = 0,

may touch the line a = O 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.