. The principles of projective geometry applied to the straight line and conic . -ough a second point. (Two solutions.) 114. Constructions involving pole and polar, and conjugate points. (a) To construct a conic such that a given point may he the pole of a given linewith respect to it and such that it (1) passes through three given points, (2) passesthrough two given points and touches a given line, (3) passes through one given point andtouches two given lines, or (4) touches three given lines. (1) Let P be the given point, p its polar,and A, B, C the given points on the conic. JoinP to A and
. The principles of projective geometry applied to the straight line and conic . -ough a second point. (Two solutions.) 114. Constructions involving pole and polar, and conjugate points. (a) To construct a conic such that a given point may he the pole of a given linewith respect to it and such that it (1) passes through three given points, (2) passesthrough two given points and touches a given line, (3) passes through one given point andtouches two given lines, or (4) touches three given lines. (1) Let P be the given point, p its polar,and A, B, C the given points on the conic. JoinP to A and B to meet p in A^ and B^. Take Aand B the harmonic conjugates of A and B withregard to PAi and PB^. The conic throughABGAB is the required conic. (2) If the conic touches a line c instead ofpassing through C. Proceed as in (1) but describea conic through ABAB to touch c. (3) and (4) are the correlatives respectively of(2) and (1). (1) and (4) have one solution while (2) and (3) have two solutions, (6) To construct a conic s\ich that a given triangle may he self-conjugate with. respect to it and such that it (1) passes through two given points, (2)a given point and touches a given line, or (3) touches two given lines. (1) Let EFG be the given triangle and Aand B the given points. Join A to E, F, G tomeet the opposite sides in Ai, A^, A^; takeA, A, A the harmonic conjugates of A withrespect to ^i ^, AiPjAs G. The conic throughA, A, A, A and B is the required conic. (2) Proceed as in (1) but describe aconic through A, A, A, A to touch the givenline h. (3) is the correlative of (1).(1) and (3) have one solution while (2) has two. through
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