. The principles of projective geometry applied to the straight line and conic . ndi^les of Projective Geometry If a rectangular hyperbola passes through the vertices of a triangle it also passesthrough the orthocentre. Let .•lZ?(^be the triangle and let the given rectangularhyperbola meet A A the perpendicular to BC in 0. Thenthrough A,B, C, C there are two rectangular hyperbolas,viz. the given one and that made up of the lines A0\BC. Therefore every conic through these points is arectangular hyperbola. Therefore BO and AC whichconstitute a conic through these points are a rectangularhyperbol
. The principles of projective geometry applied to the straight line and conic . ndi^les of Projective Geometry If a rectangular hyperbola passes through the vertices of a triangle it also passesthrough the orthocentre. Let .•lZ?(^be the triangle and let the given rectangularhyperbola meet A A the perpendicular to BC in 0. Thenthrough A,B, C, C there are two rectangular hyperbolas,viz. the given one and that made up of the lines A0\BC. Therefore every conic through these points is arectangular hyperbola. Therefore BO and AC whichconstitute a conic through these points are a rectangularhyperbola and are at right angles. Therefore 0 is theorthocentre of the triangle. A system of rectangular hyperbolas, described irith any tu-o points S and S for theends of a diameter, determines the same involution on the perpendicular diameter. Let C be the middle point of *S*S. Then, if the generating pencil be rotatedthrough an angle a the direction of one of the asymptotes is inclined at an angle , to CP, the perpendicular diameter through C. Hence the diameter CQ conjugate to. CF is inclined at an angle — - a to CS. Let Q and Q be the points in which this diameter meets thehyperbola. Let SQ apd SQ meet CP inK and K. Then, since QQ passes throughthe pole of CP, K and A are a pair ofconjugate points of the involution determinedon CP by the rectangular hyperbola. In the figure if CSQ be 6, then CSQ ise + a. The angle A*SC= angle AaS(7. Henceangle KSQ = a. Hence a circle passes throughK, Q, C, S. Therefore the angle SQK= angleSCK=A right angle. Hence QSQ is a right angle and the circleon AA as diameter passes through S and *S. Therefore CK .CK=- CS\ The involution determined by K and Kof which C is the centre is therefore the samewhatever a may be and is therefore the samefor all rectangular hyperbolas of the system. rectan ovular
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
