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angle angular points appears asymptotes becomes called centre Chap chord circle circumscribing co-ordinates coefficients coincide condition conic conic section conjugate considered constant curve denoted described determine directrix distance draw equal equation expressed find the equation fixed point foci focus follows four points given point given straight line gives harmonic Hence hyperbola identical imaginary internal line at infinity locus meet move observed obtain opposite sides pair parabola parallel passing pencil perpendicular points of contact points of intersection polar pole positive produced projection proposition prove range reciprocal relation represented respect right angles satisfy second degree seen sides similar Similarly suppose tangents term theorem third three points touch triangle of reference values whence writing written
Page 128 - Prove that the locus of the point of intersection of the tangents at P, Q, is a straight line. Shew that this straight line passes through the intersection of the directrices of the conic sections, and that the sines of the angles which it makes with these lines are inversely proportional to the corresponding excentricities.
Page 129 - ... intersection of perpendiculars of a triangle inscribed in an equilateral hyperbola lies on the curve. (246) The tangents from any point to two confocal conies are equally inclined to each other. (247) The locus of the pole of a fixed line with regard to a series of confocal conies is a straight line. (248) On a fixed tangent to a conic are taken a fixed point A and two moveable points P, Q, such that AP, AQ subtend equal angles at a fixed point 0. From P, Q are drawn two other tangents to the...
Page 142 - A parabola touches one side of a triangle in its middle point, and the other two sides produced; prove that the perpendiculars drawn from the angular points of the triangle upon any tangent to the parabola are in harmonical progression.
Page 120 - ... 8 right angles. 10. Represent the arithmetic, geometric, and harmonic means, between two given lines geometrically. 11. The centre of the circle circumscribed about any triangle, the point of intersection of the perpendiculars let fall from the angular points of the same triangle to the opposite sides, and the point of intersection of the lines joining the angular points with the middle of the opposite sides, all lie in the same right line. 12. If four circles touch each either internally or...
Page iii - FERRERS. — A Treatise on Trilinear Co-ordinates, the Method of Reciprocal Polars, and the Theory of Projections. By the Rev. NM FERRERS, MA Second Edition.
Page 10 - The plane curve described by a point which moves in such a manner that the sum of its distances from two fixed points (the foci) remains the same in all its positions.
Page 86 - In other words, if a rectangular hyperbola be so described that each angular point of a given triangle is the pole, with respect to it, of the opposite side, it will pass through the centres of the four circles which touch the three sides of the triangle.
Page 12 - To find the co-ordinates of the point of intersection of two given straight lines. Let the equations of the lines be ax + by +c = 0 (i), and a'x + b'y + c
Page 128 - OI/On, and On is constant and na fixed point. 2. Another proof is given as a problem in The Ancient and Modern Geometry of Conies, page 122 (1881), thus, " 279. If PQ be a chord of a conic which subtends a right angle at a given point...
Page 182 - If a triangle is self-conjugate with respect to each of a series of parabolas, the lines joining the middle points of its sides will be tangents: all the directrices will pass through 0 the centre of the circumscribing circle : and the focal chords, which are the polars of 0, will envelope an ellipse inscribed in the given triangle which has the nine-points' circle for its auxiliary circle.