# Mathematical Questions and Solutions Volume 25

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1876 edition. Excerpt: ... radius parallel to CA in D, and DE be drawn parallel to OA; also if OB be joined, and OQ be drawn perpendicular thereto; prove that the tangent BP, and the lines DE, OQ all pass through the same point. 4772. (By Prof. Wolstenholme.)--A circle through the foci B, C of a rectangular hyperbola meets the curve in the point Q', the tangent at Q' to the circle meets BC in O, and OL is another tangent to the circle: prove that (1) the locus of L is the (Bernoulli's) Lemniscate whose foci are B, C, and that OQ' is parallel to the straight line bisecting the angle BLC; (2) if OPQ be drawn at right angles to OQ' meeting the circle in P, Q, the locus of P will be a circular cubic of which B, C are two foci, and a third A divides BC in the ratio--1: + 1 (B being the nearer point to O); also (3) (v/2-l)CP-(v/2 + l)BP = 2AP, or AB. CP + AC. BP = 5 -AP; and (4) the angle QXQ' exceeds the angle Q'LP by a right anjle. Solution by the Proposeb.. In the Solutions to Question 4530 (Reprint, Vol. XXII., pp. 108--111) it has been shown that if A, B, C be three points on a circle, and P a fourth point on the circle, such that PA2 = PB. PC, there are four positions of P: one on the arc CA, (P); one on the arc AB, (Q); and two on the arc BC, (P', Q'). Also the straight lines PQ, P Q' are parallel to the external and internal bisectors of the angle A, and meet BC in the same point as the tangent at A to the circle; and P', Q' will be real only when a-ibc. Hence P', Q' will coincide when a-= ibc; and it is in this case that, investigating the loci of P, Q, Q' when B, C are given points, we obtain the results of the present Question 4772. Using L instead of A for the variable vertex of the triangle, the equation 3369. (Proposed by J. J. Walker, M.A.)--The...