The co-ordinates then of the point of contact are 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) 2 m2 sin2C+n sin2 B+2mn sin B sin C=n' sin'A+ sin' C = +2nl sin A sin C-l sin' B+ m2 sin*A+2lm sin A sin B, or m sin C+ n sin B=± ( sin 4+Z sinC) = ± ( 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 Cm sin C+ n (sin A - sin B) = 0 ; 7 sin B+m (sin A − sin C) – n sin B = 0. The solution of which gives (see Art. 124), 7 = 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, cos2C, and the equation of the corresponding circle, which is the inscribed circle, is cos§A √(α) + cos 1B √(B) + cos C √(y) = 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 a2 = ẞy, ß'2 = ya, y'2 = aß, and it is easy to see that A'90 A, &c. Substituting these values, the equation of the circle becomes, as before, cos A √(a) + cos 1B (ß) + cos C √(y) = 0. or a2 cos11A+ ß2 cos1 B+ y2 cos*C - 2aß cos2 A cos21⁄2B - 2ẞy cos2B cos2C - 2ya cos2 С cos2 14 = 0. We may verify that this equation represents a circle by writing it in the form y cost) (a sin 4 + ẞ sin B+ y sin C) a cos1A B cost B + + sin A sin B C 4 cos4 cos B cos* sin A sin B sin C sin C (By sin A + ya sin B + aß sin C) = 0. In the same way, the equation of one of the exscribed circles is found to be a2 cos11⁄2A +ẞ2 sin* 1⁄2 B+ y2 sin* 1 C – 2ẞy sin31⁄2B sin2 C or +2ya sin2C cos2 1⁄2A +2aß sin2†B cos2 § A = 0, cos A √√(− 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 2 cos24 cos2B cos21C (a cos A + B cos B + y cos C'} Divide by 2 cos A cos 1B cos C, and the coefficient of a in this equation is or sin C and it appears from the condition of Art. 130, that this line touches the inscribed circle the co-ordinates 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 co-ordinates 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.* * Mr. 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 132. If the equation of a circle in trilinear co-ordinates is equivalent to an equation in rectangular co-ordinates, in which the coefficient of x2+ y2 is m, then the result of substituting in the equation the co-ordinates of any point is m times the square of the tangent from that point. This constant m is easily determined in practice if there be any point, the square of the tangent from which is known by geometrical considerations; and then the length of the tangent from any other point may be inferred. Also, if we have determined this constant m for two circles, and if we subtract, one from the other, the equations divided respectively by m and m', the difference which must represent the radical axis, will always be divisible by a sinA + ẞ sin B + y sin C. Ex. 1. Find the value of the constant m for the circle through the middle points of the sides, a2 sin A cos A+ ß2 sin B cos B + y2 sin C cos C - By sin A γα sin Baß sin C = 0. Since the circle cuts any side y at points whose distances from the vertex A, are c and b cos A, the square of the tangent from A is bc cos A. But since for A we have ß=0, y=0, the result of substituting in the equation the co-ordinates of A is a22 sin A cos A, (where a' is the perpendicular from A on the opposite side), or is bc sin A sin B sin C cos A. It follows that the constant m is 2 sin A sin B sin C. Ex. 2. Find the constant m for the circle ẞy sin A + ya sin B + aß sin C. If from the preceding equation we subtract the linear terms (a cos A+B cos B + y cosC) (a sin A + ẞ sin B + y sin C), the coefficient of x2 + y2 is unaltered. The constant therefore for ẞy sin A, &c. is - sin A sin B sin C. Ex. 3. To find the distance between the centres of the inscribed and circumscribing circle. We find D2 - R2, the square of the tangent from the centre of the inscribed to 72 (sin A+ sin B+ sin C) the circumscribing circle, by substituting a=ß=y=r, to be sin A sin B sin C Ex. 4. Find the distance between the centres of the inscribed circle and of that through the middle points of sides. If the radius of the latter bep, making use of the formula, sin A cos A + sin B cos B + sin C cos C 2 sin A sin B sin C, 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. Assuming then that we otherwise know R= 2p, we have D=r-p; or the circles touch. Ex. 5. Find the constant m for the equation of the inscribed circle given above. Ans. 4 cos24 cos2B cos2C. Ex. 6. Find the tangential equation of a circle whose centre is a'ß'y' and radius r. This is investigated as in Art. 86, Ex. 4; attending to the formula of Art. 61; and is found to be 2νλ cos B - 2λμ cos('). (λa' + μß' + vy')2 = p2 (\2 + μ2 + v2 – 2μv cos A The corresponding equation in a, ß, y is deduced from this by the method afterwards explained, Art. 285, and is - p2 (a sin A+ ẞ sin B + y sin C)2 = (By' — ẞ'y)2 + (ya' — y'a)2 + (aß' — a'ß)2 -2(ya'—y'a) (aß' — a′ß) cos A −2 (aß'— a'ß)(By'—ß′y) cos B-2 (By'-ẞ'y)(ya'—y'a) cosC. This equation also gives an expression for the distance between any two points. Ex. 7. The feet of the perpendiculars on the sides of the triangle of reference from 1 1 1 (see Art. 55) lie on the same circle. By the help of the points a', B', Ex. 6, p. 60, its equation is found to be (By sinA+ya sin B+aß sin C) (a' sin A+ß′ sin B+y' sin C)(B'y' sin A+y'a' sin B+a'ß' sinC') = sin A sin B sin C (a sin A + ẞ sin B + γ sin C) (aa'(B′+y'cosA)(y′+ß'cosA) ̧ ßß'(y'+a'cosB') (a'+y'cosB) ̧ yy′(a’+ß'cosC') (B′+a′cos C')) + sin A sin B + sin C DETERMINANT NOTATION. 132(a). In the earlier editions of this book I did not venture to introduce the determinant notation, and in the preceding pages I have not supposed the reader to be acquainted with it. But the knowledge of determinants has become so much more common now than it was, that there seems now no reason for excluding the notation, at least from the less elementary chapters of the book. Thus the double area of a triangle (Art. 36), and the condition (Art. 38) that three lines should meet in a point, may be written respectively The equations of the circle through three points (Art 94), and of the circle cutting three at right angles (Ex. 2, p. 102), may be written respectively x2+ y2, x, y, 1 c', c", !!! g', -f',.1 -g", -f", 1 —g", —ƒ""', 1 The equation of the latter circle may also be formed by the help of the principle (Ex. 6, Art. 102), as the locus of the point whose polars with respect to three given circles meet in a point, in the form The corresponding equation for any three curves of the second degree will be discussed hereafter. Ex. 1. To find the condition for the co-existence of the equations ax + by + c = ax + b'y + c′ = a′′x + b′′y + c′′ = ax + b'"'y+c""'.' Let the common value of these quantities be λ; then eliminating x, y, λ from the four equations of the form ax + by + c =λ, we have the result in the form of a determinant 1, 1, 1, 1 or A+ C = B+ D, where A, B, C, D are the four minors got by erasing in turn each column, and the top row in this determinant. To find the condition that four lines should touch the same circle, is the same as to find the condition for the co-existence of the equations aß = y=d. In this case the determinants A, B, C, D geometrically represent the product of each side of the quadrilateral formed by the four lines, by the sines of the two adjacent angles. Ex. 2. To find the relation connecting the mutual distances of four points on a circle. The investigation is Mr. Cayley's (see Lessons on Higher Algebra, p. 21). Multiply together according to the ordinary rule the determinants which are only different ways of writing the condition of Art. 94; and we get the required relation 0, (12)2, (13)2, (14)2 where (12)2 is the square of the distance between two points. This determinant expanded is equivalent to (12) (34) ± (13) (24) ± (14) (23) = 0. Ex. 3. To find the relation connecting the mutual distances of any four points in a plane. This investigation is also Mr. Cayley's (Lessons on Higher Algebra, p. 22). Prefix a unit and cyphers to each of the determinants in the last example; thus |