An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory of Projections |
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Page 4
... Y , of the second degree . * This , if not self - evident , may be proved as follows : Let P , Qle the two given points . Join PQ , and draw PM , QM ' perper . DISTANCE BETWEEN TWO POINTS . 5 Again , since - 4 TRILINEAR CO - ORDINATES .
... Y , of the second degree . * This , if not self - evident , may be proved as follows : Let P , Qle the two given points . Join PQ , and draw PM , QM ' perper . DISTANCE BETWEEN TWO POINTS . 5 Again , since - 4 TRILINEAR CO - ORDINATES .
Page 9
... follows . Let Q be any point on the line , a , ß , y its co - ordinates . Draw QK perpendicular to AC , QL to AB . Then , as before , we have QK = QL . It will however be observed , that if Q and B lie on the same side of AC , Q and C ...
... follows . Let Q be any point on the line , a , ß , y its co - ordinates . Draw QK perpendicular to AC , QL to AB . Then , as before , we have QK = QL . It will however be observed , that if Q and B lie on the same side of AC , Q and C ...
Page 23
... follow one another . Thus , the an- harmonic ratio of the pencil OP , OR , OQ , OS , is different from that of the pencil OP , OQ , OR , OS , the former being equal to sin POR . sin QOS sin POQ . sin ROS . sin POS . sin QOR ' the latter ...
... follow one another . Thus , the an- harmonic ratio of the pencil OP , OR , OQ , OS , is different from that of the pencil OP , OQ , OR , OS , the former being equal to sin POR . sin QOS sin POQ . sin ROS . sin POS . sin QOR ' the latter ...
Page 27
... follows that the straight lines respectively represented by the equations B = 0 , B − ky = 0 , y = 0 , B + ky = 0 , form an harmonic pencil . 24. Hence we deduce a geometrical construction for the determination of the fourth harmonic ...
... follows that the straight lines respectively represented by the equations B = 0 , B − ky = 0 , y = 0 , B + ky = 0 , form an harmonic pencil . 24. Hence we deduce a geometrical construction for the determination of the fourth harmonic ...
Page 31
... follows . Let p , p , q , q ' be the respective distances of the four points from any arbitrary point on the line , x the distance of the centre from the same point . Then , by definition , ( p − x ) ( p ' — x ) = ( q − x ) ( q ′ — x ) ...
... follows . Let p , p , q , q ' be the respective distances of the four points from any arbitrary point on the line , x the distance of the centre from the same point . Then , by definition , ( p − x ) ( p ' — x ) = ( q − x ) ( q ′ — x ) ...
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Common terms and phrases
angular points anharmonic ratio asymptotes ax+bB+cy b₁ c₁ centre chord circumscribing common tangents condition of tangency conic section conics intersect described determine directrix distance equation escribed circles find the equation fixed point foci focus four points given conic given point given straight line given triangle harmonic pencil Hence imaginary internal bisectors investigated Let the equation line at infinity lines joining locus meets the conic nine-point circle pair parallel Pascal's Theorem perpendicular points at infinity points of contact points of intersection polar reciprocal pole PQRS projection prove radical axis reciprocal polars reciprocated with respect rectangular hyperbola represented right angles second degree self-conjugate shew sin POS system of conics tangents drawn theorem three points three straight lines touches the line triangle of reference TRILINEAR CO-ORDINATES Ua² ux² vß² wy² λα
Popular passages
Page 110 - ... 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 118 - 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 xiv - 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 76 - 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 2 - 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 118 - 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 160 - Any straight line drawn from the vertex of a triangle to the base is bisected by the straight line which joins the middle points of the other sides of the triangle.
Page 114 - Let them be denoted by F and F' (fig. 72), and let the axis of x be taken through them, and the origin halfway between them. Then if P is any point on the ellipse and 2 a represents the constant sum of its distances from the foci, we have F'P+FP=2a.
Page 119 - ... subtends a right angle at a fixed point. Prove that the locus of the point of intersection of the variable tangents is a straight line.