Writing the equations of the tangents at the three vertices

we see that the three points in which each intersects the opposite side are in one right line, whose equation is

Subtracting, one from another, the equations of the three tangents, we get the equations of the lines joining the vertices of the original triangle to the corresponding vertices of the triangle formed by the three tangents, viz.,

three lines which meet in a point (Art. 40).*

127. If a'B'y', a"B"y" be the coordinates of any two points on the curve, the equation of the line joining them is

for if we substitute in this equation a'B'y' for aßy, the equation is satisfied, since a"B"y" satisfy the equation of the curve, which may be written

n

7 m
+ + = 0.
B ช

In like manner the equation is satisfied by the coordinates a"B"y". It follows that the equation of the tangent at any point a'B'y' may be written.

and conversely, that if λa+μß+vy = 0 is the equation of a tangent, the coordinates of the point of contact a'B'y' are given by the equations

The theorems of this article are by M. Bobillier (Annales des Mathématiques, vol. xvIII. p. 320). The first equation of the next article is by M. Hermes.

Solving for a', ', y' from these equations, and substituting in the equation of the curve, which must be satisfied by the point a'B'y', we get

V(r) + V(mu)+ V(nv)=0.

This is the condition that the line λa+μß+vy may touch By + mya+naß; or it may be called (see Art. 70) the tangential equation of the curve. The tangential equation might also be obtained by eliminating y between the equation of the line and that of the curve, and forming the condition that the resulting equation in a : ẞ may have equal roots.

128. To find the conditions that the general equation of the second degree in ́a, B, Y,

aa* +bB+cy*+2fBy+2gya + 2haß = 0,

may represent a circle. [Dublin Exam. Papers, Jan. 1857].

It is convenient to avail ourselves of the result of Art. 124. Since the terms of the second degree, x2+ y2, are the same in the equations of all circles, the equations of two circles can only differ in the linear part; and if S represent a circle, an equation of the form S+ lx+my+n=0 may represent any circle whatever. In like manner, in trilinear coordinates, if we have found one equation which represents a circle, we have only to add to it terms la+mẞ+ny (which in order that the equation may be homogeneous we multiply by the constant a sin4+ẞsin B+ysin C), and we shall have an equation which may represent any circle whatever. Thus then (Art. 124) the equation of any circle may be thrown into the form

(la+mẞ+ny) (a sin A+ ẞ sin B+ y sin C)

+k(By sin A+ ya sin B+ aß sin C′) = 0. If now we compare the coefficients of a2, B2, y" in this form with those in the general equation, we see that, if the latter represent a circle, it must be reducible to the form

B+ y) (a sin 4 + ẞ sin B + y sin C)

sin C

+k (By sin A + ya sin B+ aß sin C) = 0

R

and a comparison of the remaining coefficients gives

2f sin B sin C=c sin B+ b sin C+ k sin A sin B sin C, 2g sin C sin A = a sin2 C+ c sin3A + k sin A sin B sin C, 2h sin A sin B = b sin*A + a sin*B + k sin A sin B sin C, whence eliminating k, we have the required conditions, viz. b sin C+ c sin B-2f sin B sin C = c sin3A + a sin3 C-2g sin C sinA = a sin B+b sin'A – 2h sin A sin B.

If we have the equations of two circles written in the form (la + mẞ + ny) (a sin A + B sin B+ y sin C)

+(By sin A+ ya sin B+ aß sin C)=0,

(l'a + m'ß + n'y) (a sin A + B sin B+ y sin C)

+k (By sin A+ ya sin B + aß sin C) = 0,

it is evident that their radical axis is

la + mß + ny — (l'a + m'ß + n'y),

and that la+mẞ+ny is the radical axis of the first with the circumscribing circle.

Ex. 1. Verify that aß - y2 represents a circle if A = B (Art. 123).

The equation may be written

aß sin C+ By sin A+ ya sin B − y (a sin A + ẞ sin B + y sin C') = 0.

Ex. 2. When will aa2 + bẞ2+ cy2 represent a circle?

Ex. 3. The three middle points of sides, and the three feet of perpendiculars lie on a circle. The equation

a2 sin A cos 4 + ß2 sin B cos B + y2 sin C cos C- (ẞy sin A + ya sin B + aß sin C)=0, represents a curve of the second degree passing through the points in question. For if we make y = 0, we get

a2 sin A cos A + B2 sin B cos B - aß (sin A cos B + sin B cos A) = 0, the factors of which are a sin A - ẞ sin B and a cos A a circle, for it may be written

-

(a cos A+ ẞ cos B + y cos C) (a sin A + ẞ sin B + y sin C)

B cos B. Now the curve is

2 (By sin A+ ya sin B+ aß sin C)=0 Thus the radical axis of the circumscribing circle and of the circle through the middle points of sides is a cos A+ẞ cos B + y cos C, that is, the axis of homology of the given triangle with the triangle formed by joining the feet of perpendiculars.

129. We shall next show how to form the equations of the circles which touch the three sides of the triangle a, B, y. The

general equation of a curve of the second degree touching the three sides is

l'a2 + m2ß2 + n3y3 — 2mnßy — 2nlya — 2lmaß = 0.*

Thus y is a tangent, or meets the curve in two coincident points, since, if we make y=0 in the equation, we get the perfect square l'a2 + m2ß2 – 2lmaß = 0. The equation may also be written in a convenient form

√(la) + √(mB) + √(ny) = 0;

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

Before determining the values of l, m, n, for which the equation represents a circle, we shall draw from it some inferences. which apply to all curves of the second degree inscribed in the triangle. Writing the equation in the form

ny (ny - 2la - 2mß) + (la — mß)2 = 0,

we see 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 are

la-m3-0, mẞ-ny=0, ny - la=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— mB)* = 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

* Strictly speaking, the double rectangles in this equation ought to be written with the ambiguous sign ±, and the argument in the text would apply equally. If, hiowever, we give all the rectangles positive signs, or if we give one of them a positive sign, and the other two negative, the equation does not denote a proper curve of the second degree, but the square of some one of the lines la mẞny. And the form in the text may be considered to include the case where one of the rectangles is negative and the other two positive, if we suppose that l, m, or n may denote a negative as well as a positive quantity.

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+2m3-ny =0, 2mB+2ny - la=0, 2ny+2la – mB=0. 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 7, of the second with a, and of the third with B.

130. The equation of the chord joining two points a'B'y', a"B"y", on the curve is

a √(?) {√(B'y'') + √(B′′y')} +B √(m) {√(y ́a′′) + √(y′′a')}

+ y√(n) {√(a ́'B'') + √(a′′B′)} = 0.*

For substitute a', B', y for a, B, y, and it will be found that the quantity on the left-hand side may be written.

{√(á ́B ́y′′) + √(B'ý ́a'') + √(y ́a' ́ß'')} {√(la') + √(mB') + √√(ny)}

— √(a ́B ́y') {√(la′′) + √(mB') + √ (ny'')},

which vanishes, since the points are on the curve. The equation of the tangent is found by putting a′′, B′′, y′′ = d′, B', y′ in the above. Dividing by 2 (d'B'y), it becomes

Conversely, if λa + μß + vy is a tangent, the coordinates of the point of contact are given by the equations

Solving for a'B'y', and substituting in the equation of the curve, we get

+ = λ μ ν

0,

which is the condition that λa+μB+vy may be a tangent; that is to say, is the tangential equation of the curve.

*This equation is Dr. Hart's.