In the latter theorem, if the fixed line be at infinity, we find the envelope of the asymptotes of a series of hyperbolas, having the same focus and same directrix, to be a parabola having the same focus and touching the common directrix. If two chords at right angles to each other be drawn through any point on a circle, the line joining their extremities passes through the centre. The locus of the intersection of tangents to a parabola which cut at right angles is the directrix. We say a parabola, for, the point through which the chords of the circle are drawn being taken for origin, the polar of the circle is a parabola (Art. 308). The envelope of a chord of a circle which subtends a given angle at a given point on the curve is a concentric circle. Given base and vertical angle of a triangle, the locus of vertex is a circle passing through the extremities of the base. The locus of the intersection of tangents to an ellipse or hyperbola which cut at right angles is a circle. The locus of the intersection of tangents to a parabola, which cut at a given angle, is a conic having the same focus and the same directrix. Given in position two sides of a triangle, and the angle subtended by the base at a given point, the envelope of the base is a conic, of which that point is a focus, and to which the two given sides will be tangents. The envelope of any chord of a conic which subtends a right angle at any fixed point is a conic, of which that point is a focus. "If from any point on the circumference of a circle perpendiculars be let fall on the sides of any inscribed triangle, their three feet will lie in one right line" (Art. 125). If we take the fixed point for origin, to the triangle inscribed in a circle will correspond a triangle circumscribed about a parabola; again, to the foot of the perpendicular on any line corresponds a line through the corresponding point perpendicular to the radius vector from the origin. Hence, "If we join the focus to each vertex of a triangle circumscribed about a parabola, and erect perpendiculars at the vertices to the joining lines, those perpendiculars will pass through the same point." If, therefore, a circle be described, having for diameter the radius vector from the focus to this point, it will pass through the vertices of the circumscribed triangle. Hence, Given three tangents to a parabola, the locus of the focus is the circumscribing circle (p. 207). The locus of the foot of the perpendicular (or of a line making a constant angle with the tangent) from the focus If from any point a radius vector be drawn to a circle, the envelope of a perpendicular to it at its extremity (or of a of an ellipse or hyperbola on the tangent line making a constant angle with it) is a is a circle conic having the fixed point for its focus. 310. Having sufficiently exemplified in the last Article the method of transforming theorems involving angles, we proceed to show that theorems involving the magnitude of lines passing through the origin are easily transformed by the help of the first theorem in Art. 307. For example, the sum (or, in some cases, the difference, if the origin be without the circle) of the perpendiculars let fall from the origin on any pair of parallel tangents to a circle is constant, and equal to the diameter of the circle. Now, to two parallel lines correspond two points on a line passing through the origin. Hence, "the sum of the reciprocals of the segments of any focal chord of an ellipse is constant." We know (p. 185) that this sum is four times the reciprocal of the parameter of the ellipse, and since we learn from the present example that it only depends on the diameter, and not on the position of the reciprocal circle, we infer that the reciprocals of equal circles, with regard to any origin, have the same parameter. The rectangle under the segments of any chord of a circle through the origin is constant. The rectangle under the perpendiculars let fall from the focus on two parallel tangents is constant. Hence, given the tangent from the origin to a circle, we are given the conjugate axis of the reciprocal hyperbola. Again, the theorem that the sum of the focal distances of any point on an ellipse is constant may be expressed thus: The sum of the distances from the focus of the points of contact of parallel tangents is constant. The sum of the reciprocals of perpendiculars let fall from any point within a circle on two tangents, whose chord of contact passes through the point, is constant. 311. If we are given any homogeneous equation connecting the perpendiculars PA, PB, &c. let fall from a variable point P on fixed lines, we can transform it so as to obtain a relation connecting the perpendiculars ap, bp' &c., let fall from the fixed points a, b, &c., which correspond to the fixed lines, on the variable line which corresponds to P. For we have only to divide the equation by a power of OP, the distance of P from the origin, and then, by Art. 101, substitute for each term РА ар OP' Oa For example, if PA, PB, PC, PD be the perpendiculars let fall from any point of a conic on the sides of an inscribed quadrilateral, PA.PC=kPB.PD (Art. 259). Dividing each factor by OP, and substituting, as above, we have áp áp bp' do"" Ob Od; and Oa, Ob, Oc, Od being constant, we Oa Oc =k infer that if a fixed quadrilateral be circumscribed to a conic, the product of the perpendiculars let fall from two opposite vertices on any variable tangent is in a constant ratio to the product of the perpendiculars let fall from the other two vertices. The product of the perpendiculars from any point of a conic on two fixed tangents is in a constant ratio to the square of the perpendicular on their chord of contact. (Art. 259). The product of the perpendiculars from two fixed points of a conic on any tangent, is in a constant ratio to the square of the perpendicular on it, from the intersection of tangents at those points. If, however, the origin be taken on the chord of contact, the reciprocal theorem is "the intercepts, made by any variable tangent on two parallel tangents, have a constant rectangle." The product of the perpendiculars on any tangent of a conic from two fixed points (the foci) is constant. The square of the radius vector from a fixed point to any point on a conic, is in a constant ratio to the product of the perpendiculars let fall from that point of the conic on two fixed right lines. Generally, since every equation in trilinear coordinates is a homogeneous relation between the perpendiculars from a point on three fixed lines, we can transform it by the method of this article, so as to obtain a relation connecting λ, u, v, the perpendiculars let fall from three fixed points on any tangent to the reciprocal curve, which may be regarded as a kind of tangential equation* of that curve. Thus the general trilinear equation of a conic becomes, when transformed, where p, p, p" are the distances of the origin from the vertices of the new triangle of reference. Or, conversely, if we are given any relation of the second degree Ax2+ &c. =0, con * See Appendix on Tangential Equations. necting the three perpendiculars λ, u, v, the trilinear equation of the reciprocal curve is a2 B2 By +2F +2G + 2H = αβ α' β' 0, where a', B', y' are the trilinear coordinates of the origin. Ex. 1. Given the focus and a triangle circumscribing a conic, the perpendiculars let fall from its vertices on any tangent to the conic are connected by the relation where 0, 0, 0" are the angles the sides of the triangle subtend at the focus. This is obtained by forming the reciprocal of the trilinear equation of the circle circumscribing a triangle. If the centre of the inscribed circle be taken as focus, we have 0 = 90° + 4, p sin 4 = r, whence the tangential equation, on this system, of the inscribed circle is μv cot Avλ cot B +λu cot C = 0. In the case of any of the exscribed circles two of the cotangents are replaced by tangents. Ex. 2. Given the focus and a triangle inscribed in a conic, the perpendiculars let fall from its vertices on any tangent are connected by the relation sin √() + sin 40' (~) + sin 30" |(~~) = 0. The tangential equation of the circumscribing circle takes the form sin A √() + sin B √(u) + sin C √(v) = 0. Ex. 3. Given focus and three tangents the trilinear equation of the conic is This is obtained by reciprocating the equation of the circumscribing circle last found. Ex. 4. In like manner, from Ex. 1, we find that given focus and three points the trilinear equation is tan 10 +tan 10 B +tan 10" = 0. 312. Very many theorems concerning magnitude may be reduced to theorems concerning lines cut harmonically or anharmonically, and are transformed by the following principle: To any four points on a right line correspond four lines passing through a point, and the anharmonic ratio of this pencil is the same as that of the four points. This is evident, since each leg of the pencil drawn from the origin to the given points is perpendicular to one of the corresponding lines. We may thus derive the anharmonic properties of conics in general from those of the circle. The anharmonic ratio of the pencil joining four points on a conic to a variable fifth is constant. The anharmonic ratio of the point in which four fixed tangents to a conic cut any fifth variable tangent is constant. The first of these theorems is true for the circle, since all the angles of the pencil are constant, therefore the second is true for all conics. The second theorem is true for the circle, since the angles which the four points subtend at the centre are constant, therefore the first theorem is true for all conics. By observing the angles which correspond in the reciprocal figure to the angles which are constant in the case of the circle, the student will perceive that the angles which the four points of the variable tangent subtend at either focus are constant, and that the angles are constant which are subtended at the focus by the four points in which any inscribed pencil meets the directrix. 313. The anharmonic ratio of a line is not the only relation concerning the magnitude of lines which can be expressed in terms of the angles subtended by the lines at a fixed point. For, if there be any relation which, by substituting (as in Art. 56) OA. OB.sin A OB for each line AB involved in it, ОР can be re duced to a relation between the sines of angles subtended at a given point O, this relation will be equally true for any transversal cutting the lines joining O to the points A, B, &c.; and by taking the given point for origin a reciprocal theorem can be easily obtained. For example, the following theorem, due to Carnot, is an immediate consequence of Art. 148: "If any conic meet the side AB of any triangle in the points c, c'; BC in a, a'; AC in b, b'; then the ratio Now, it will be seen that this ratio is such that we may substitute for each line Ac the sine of the angle AOc, which it subtends at any fixed point; and if we take the reciprocal of this theorem, we obtain the theorem given already Art. 295. 314. Having shown how to form the reciprocals of particular theorems, we shall add some general considerations respecting reciprocal conics. We proved (Art. 308) that the reciprocal of a circle is an ellipse, hyperbola, or parabola, according as the origin is within, PP. |