. The principles of projective geometry applied to the straight line and conic . l whenthe triangle is obtuse angled. Let B, E, F be the feet of the perpen-diculars from the vertices on the oppositesides, and 0 the orthocentrc of the triangleABC. Then OE. OB = OC. 0F= OB. OA = K- suppose. Hence BC, CA, AB are thepolars of A, B, C with respect to acircle centre 0 and radius K. This is therequired circle. If the triangle is acute angled OE. OBis negative and the circle is imaginary but its centre, the orthocentre of the triangle,is real. There is only one circle with regard to which a given tria
. The principles of projective geometry applied to the straight line and conic . l whenthe triangle is obtuse angled. Let B, E, F be the feet of the perpen-diculars from the vertices on the oppositesides, and 0 the orthocentrc of the triangleABC. Then OE. OB = OC. 0F= OB. OA = K- suppose. Hence BC, CA, AB are thepolars of A, B, C with respect to acircle centre 0 and radius K. This is therequired circle. If the triangle is acute angled OE. OBis negative and the circle is imaginary but its centre, the orthocentre of the triangle,is real. There is only one circle with regard to which a given triangle is self-conjugate. (6) The pedal triangle of ABC is BEF. ABC is self-polar with regard toa circle A. Prove that the polars of DEF with respect to the circle X form atriangle similar to ABC and of twice its dimensions. F and C are conjugate points with regard to the circle A, therefore the polarof F is a line through C perpendicular to FC, that is, parallel to AB. Hence thepolar triangle of DEF is the triangle formed by drawing through ABC linesparallel to opposite 156 Principles of Projective Geometry (7) Show that the square of the distance between a pair of conjugate pointswith respect to a circle is equal to the sum of thepowers of the points. Let the polar of A meet the circle in L and B any point on LAf. Join A to 0 the centreof circle 0 to meet LM in N. Then NM-^
Size: 1682px × 1486px
Photo credit: © Reading Room 2020 / Alamy / Afripics
License: Licensed
Model Released: No
Keywords: ., bookcentury1900, bookdecade1910, booksubjectgeometryprojective
