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

l2a2 + m2ß2 + n3y2 – 2mnẞy — 2nlya — 2lmaß = 0.*

Thus γ 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'a3 + m2ß2 – 2lmaß = 0. l'a2+m*B* - 2lmaß=0. The equation may also

be written in a convenient form

√(la) + √(mß) + √√(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 - mß)2 = 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 mẞ±ny. And the form in the text may be considered to include the case where one of the rectanglesi 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 + 2mB — ny =0, 2mB+2ny - la=0, 2ny +2la – mß = 0.

Hence we infer that the three points, where each of these tangents meets the opposite side, lie in one right line,

la + mB+ny = 0,

for this line passes through the intersection of the first line with of the second with a, and of the third with ß.

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′8′′) + √√(a′′B')} = 0.*

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

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

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

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

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

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

7 m
+ +

n

λ μ V

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

This equation is Dr. Hart's.

The reciprocity of tangential and ordinary equations will be better seen if we solve the converse problem, viz. to find the equation of the curve, the tangents to which fulfil the condition.

"a+μ"B+v"y be any two lines, such that X'u'v', ""v" satisfy the above condition, and which therefore are tangents to the curve whose equation we are seeking; then

is the tangential equation of their point of intersection. For (Art. 70) any equation of the form A+ Bu + Cv=0 is the condition that the line λα + β + my should pass through a certain point, or, in other words, is the tangential equation of a point; and the equation we have written being satisfied by the tangential coordinates of the two lines is the equation of their point of intersection. Making X', u', v = X", μ", v′′ we learn that if there be two consecutive tangents to the curve, the equation of their point of intersection, or, in other words, of their point of contact, is

a mp ην
λ μ

+ + =

Solving for X', u', v' from these equations, and substituting in the relation, which by hypothesis X'u'v' satisfy, we get the required equation of the curve

√(la) + √(mß) + √√(ny) = 0.

131. The conditions that the equation of Art. 129 should represent a circle are (Art. 128)

m2 sin C+n" sin'B+2mn sin B sin C-n' sin'A+ sin" C

=

+2nl sin C sin Al sin' B+ m2 sin*A+2lm sin A sin B, or m sin C+n sin B=+ (n sin A + 7 sin C) = ± (l sin B+m sin A).

Four circles then may be described to touch the sides of the given triangle, since, by varying the sign, these equations may be written in four different ways. If we choose in both cases the sign, the equations are

7 sin C-m sin C+ n (sin A - sin B) = 0;

I sin B+m (sin A − sin C) – n sin B = 0.

The solution of which gives (see Art. 124)

1 = sin A (sin B+ sin C-sin A), m = sin B (sin C+ sin A − sin B), n = sin C (sin A+ sin B − sin C).

But since in a plane triangle

sin B+ sin C-sin A = 4 cos A sin B sin C,

these values for l, m, n are respectively proportional to cos31⁄2A, cos B, cos C, and the equation of the corresponding circle, which is the inscribed circle, is

or

cos 4√(a) + cos B√(8) + cos C √√/ (v) = 0,*

a2 cos*4 +82 cos*}B+ y2 cos*1⁄2 C – 2ẞy cos2B cos2 C

- 2ya cos2C cos31⁄2A – 2aß cos" A cos2+B=0.

We may verify that this equation represents a circle by writing it in the form

a cos A, B cos1B sin A

(a co

+

B cos* + B + Y cos) (a sin 4 + 8 sin B + y sin C')

sin B

sin C

(By sin A + ya sin B+ a3 sin C) = 0.

* Dr. Hart derives this equation from that of the circumscribing circle as follows: Let the equations of the sides of the triangle formed by joining the points of contact of the inscribed circle be a' = 0, ß' = 0, y' = 0, and let its angles be A', B', C'; then (Art. 124) the equation of the circle is

B'y' sin A' + y'a' sin B' + a'ß' sin C' = 0.

But (Art. 123) for every point of the circle we have a'2 = ẞy, ẞ" = ya, y'2 = aß, and it is easy to see that A'90 - §4, &c. Substituting these values, the equation

of the circle becomes, as before,

cos 4 √(a) + cos B √(ẞ) + cos &C √(y) = 0.

If the equation of the note, p. 119, be treated similarly, we find that every point of the circle, of which a, ß, y, ò are tangents, satisfies the equation,

cos (12) cos (23) cos (34) cos (41)

+

+

(αβ) J(BY) 4(28)

where (12) denotes the angle between aß, &c. Similarly for any number of tangents.

In the same way, the equation of one of the exscribed circles is found to be

a2 cos*4 + ẞ2 sin*}B+ y2 sin*} C – 2ẞy sin'B sin*C

or

+2ya sin2C cos2 4+2aß sin31⁄2В cos31⁄2A = 0,

cos 4 √(-a) + sin B √(B) + sin C√(y) = 0.

The negative sign given to a is in accordance with the fact, that this circle and the inscribed circle lie on opposite sides of the

line a.

Ex. Find the radical axis of the inscribed circle and the circle through the middle points of sides.

The equation formed by the method of Art. 128 is

Divide by 2 cos A cos B cos C, and the coefficient of a in this equation is

or

cos 4 (2 cos2 A sin B sin C - cos A cos B cos &C'},

cos A sin (A — B) sin (A — C).

The equation of the radical axis then may be written

and it appears from the condition of Art. 130 that this line touches the inscribed circle, the coordinates of the point of contact being sin2 § (B-C), sin2 } (C—A), sin2 § (A — B). These values shew (Art. 66) that the point of contact lies on the line joining the two centres whose coordinates are 1, 1, 1, and cos (B — C), cos (C – A), cos (A — B). In the same way it can be proved that the circle through the middle points of sides touches all the circles which touch the sides. This theorem is due to Feuerbach.*

* Dr. Casey has given a proof of Feuerbach's theorem, which will equally prove Dr. Hart's extension of it, viz. that the circles which touch three given circles can be distributed into sets of four, all touched by the same circle. The signs in the following correspond to a triangle whose sides are in order of magnitude a, b, c. The exscribed circles are numbered 1, 2, 3, and the inscribed 4; the lengths of the direct and transverse common tangents to the first two circles are written (12), (12)'. Then because the side a is touched by the circle 1 on one side, and by the other three circles on the other, we have (see p. 115)

showing that the four circles are also touched by a circle, having the circle 4 on one side and the other three on the other.