An Elementary Treatise on Trilinear Co-ordinates: The Method of Reciprocal Polars, and the Theory Projections |
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Page 1
... positive , and that of the other as negative . We shall consider a , the distance of a point from the line BC , as positive if the point lie on the same side of that line as the point A does , negative if on the other side ; and ...
... positive , and that of the other as negative . We shall consider a , the distance of a point from the line BC , as positive if the point lie on the same side of that line as the point A does , negative if on the other side ; and ...
Page 2
... , we get aa + bB + cy = 2A . Next , suppose P to lie between AB , AC produced , and on the side of BC remote from A ( fig . 2 ) . Then a will be Fig . 2 . E negative , while B , y are positive . Hence 2 TRILINEAR CO - ORDINATES .
... , we get aa + bB + cy = 2A . Next , suppose P to lie between AB , AC produced , and on the side of BC remote from A ( fig . 2 ) . Then a will be Fig . 2 . E negative , while B , y are positive . Hence 2 TRILINEAR CO - ORDINATES .
Page 3
... positive . Hence , twice the area PBC will be represented by aa , and we shall therefore have as before - aa + bB + cy = 2A . Thirdly , let P lie between AB , AC , produced backwards ( fig . 3 ) , so that B , y are negative while a is ...
... positive . Hence , twice the area PBC will be represented by aa , and we shall therefore have as before - aa + bB + cy = 2A . Thirdly , let P lie between AB , AC , produced backwards ( fig . 3 ) , so that B , y are negative while a is ...
Page 30
... , k2 suppose . Then these points are said to form a system in involution . If K be a point such that OK2 = k2 , K is called a focus of the system . If k2 be positive , there will evidently be two 30 TRILINEAR CO - ORDINATES .
... , k2 suppose . Then these points are said to form a system in involution . If K be a point such that OK2 = k2 , K is called a focus of the system . If k2 be positive , there will evidently be two 30 TRILINEAR CO - ORDINATES .
Page 31
... positive , there will evidently be two such foci , one on each side of O , if negative ( and k therefore imagi- nary ) there will be no real foci . The point is called the centre of the system . Two points , such as P , P ' , are said ...
... positive , there will evidently be two such foci , one on each side of O , if negative ( and k therefore imagi- nary ) there will be no real foci . The point is called the centre of the system . Two points , such as P , P ' , are said ...
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An Elementary Treatise on Trilinear Coordinates: The Method of Reciprocal Norman MacLeod Ferrers No preview available - 2023 |
Common terms and phrases
aa+bB+cy angular points asymptotes auxiliary conic b₁ C₁ centre Chap chord co-ordinates coefficients common tangents conic section determinant directrix find the equation fixed point fixed straight line focus follows four points given conic given point given straight line given triangle h₂ Hence imaginary investigated Let the equation line at infinity line joining locus meets the conic nine-point circle obtain opposite sides parabola Pascal's Theorem perpendicular point f points at infinity points of contact points of intersection prove radical axis reciprocal curve reciprocated with respect rectangular hyperbola represented right angles second degree shewn similar and similarly sin POS tangents drawn theorem three points three straight lines triangle of reference ua² uc² uf+w'g+v'h V'ca va² Vb² vß² W'ab wa² Wc² whence wy² λα
Popular passages
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.