A first course in projective geometry . gonally when it is adiameter. But in the parabola the straight line midway between apole and its polar and parallel to the latter is a tangent(Chapter VIII. § 11, Cor. 2). Hence the lines joining themiddle points of the sides of a self-con jugate triangle aretangents to the parabola. Also the centre of the circumcircle of the self-conjugatetriangle is the orthocentre of the triangle formed by joiningthe middle points of the sides. .. the orthocentre of the triangle formed by three tangents IS on the directrix. Ex. From properties of the parabola prove (a
A first course in projective geometry . gonally when it is adiameter. But in the parabola the straight line midway between apole and its polar and parallel to the latter is a tangent(Chapter VIII. § 11, Cor. 2). Hence the lines joining themiddle points of the sides of a self-con jugate triangle aretangents to the parabola. Also the centre of the circumcircle of the self-conjugatetriangle is the orthocentre of the triangle formed by joiningthe middle points of the sides. .. the orthocentre of the triangle formed by three tangents IS on the directrix. Ex. From properties of the parabola prove (assuming that only oneparabola can be drawn to touch four given straight lines) that (1) the circumcircles of the four triangles formed by four given straight lines are concurrent; (2) the orthocentres of these triangles are collinear. § 12. Curvature. Consider two curves touching one another at a point the common tangent at A, and from a near point P on itdraw PQ^ at right angles to PA to cut the curves at Q and Fio. 94. Then PQ, P^ measure the deflections of the curves from thetangent line (Fig. 94). M 178 PROJECTIVE GEOMETRY PQConsider the ratio —-^. If it were unity for all positions of P near A, the curves would coincide in that neighbourhood, and their curvatures would be the same. We shall therefore not be inconsistent if we define the ratio of the curvatures of the curves at A in the general case PQ,as the limiting value of the ratio when P moves up to A. Pq Consider the circle. If any two points A, a (Fig. 95a) betaken on it and AP, aj) be the tangents, PQ, pq at right anglesto these to meet the circle in Q, q ; then if AP = ap, it is easilyproved that PQ =^:>2.
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