. The principles of projective geometry applied to the straight line and conic . abola. (1) The directrix of a parabola passes through the orthocentreof every circumscribed triangle, or The directriw of every parabolainscribed in a triangle passes through its orthocentre. Let ABG be a triangle cir-cumscribed to a parabola, 0 itsorthocentre, a, b, c its sides and00 the line at infinity. Thenabc<x) is a circumscribed quadri-lateral of the parabola. Considerthe involution subtended by itat 0. A pair of conjugate raysof this involution are the linesjoining 0 to be and aoo . Theseare parallel to
. The principles of projective geometry applied to the straight line and conic . abola. (1) The directrix of a parabola passes through the orthocentreof every circumscribed triangle, or The directriw of every parabolainscribed in a triangle passes through its orthocentre. Let ABG be a triangle cir-cumscribed to a parabola, 0 itsorthocentre, a, b, c its sides and00 the line at infinity. Thenabc<x) is a circumscribed quadri-lateral of the parabola. Considerthe involution subtended by itat 0. A pair of conjugate raysof this involution are the linesjoining 0 to be and aoo . Theseare parallel to OA and a andare therefore at right the lines joining 0 toba and coo are at right the involution pencil is orthogonal and the tangents from0 to the parabola are at right angles. Therefore 0 is on its directrix. (2) The directrix of a parabola inscribed in a quadrilateral passesthrough the orthocentre of each of the four triangles formed by the sidesof the quadrilateral and therefore the four orthocentres are collinear. This follows fi-om (1).. Deductions from Desargiies Theorem 223 (3) The circles descnbed on the diagonals of a quadrilateral asdiameters meet the line of orthocentres in the same pair of points andare therefore coaxal. Consider either of the points of intersection of two of the circles. Theinvolution subtended at this point by the vertices of the quadrilateralis orthogonal. Hence the circle described on the third diagonal passesthrough the point and it is a point on the directrix of the inscribedparabola. Hence it is on the line of orthocentres of the quadrilateral. From (8) it follows that the middle points of the diagonals of thequadrilateral, being the centres of a system of coaxal circles, arecollinear, and that their line of col linearity is perpendicular to the lineof orthocentres, which is the radical axis of the coaxal circles. (ii) Properties of the Director Circle of a Conic. (1) The director circles of a system of coni
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Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
