# Applicable Geometry (1977)(en)(207s) by Heinrich W Guggenheimer

By Heinrich W Guggenheimer

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T P , T Q are the tangents at the points P , Q of a conic, and P Q meets the directrix in R; prove that RST is a right angle. 20. SR being the semi-latus rectum, if RA meet the directrix in E, and SE meet the tangent at the vertex in T , AT = AS. 21. If from any point T , in the tangent at P , T M be drawn perpendicular to SP , and T N perpendicular to the transverse axis, meeting the curve in R, SM = SR. 22. If the chords P Q, P Q meet the directrix in F and F , the angle F SF is half P SP . 23.

55. Let S be the focus, EX the directrix, and SX the perpendicular on EX from S. Divide SX at the point A in the given ratio; the point A is the vertex. From any point E in EX, draw EAP , ESL, and through S draw SP making the angle P SL equal to LSN , and meeting EAP in P .

For the angle and T P S = 21 P ST , T P S = 12 P ST, ∴ T P T = 12 (P ST + P ST ) = a right angle. It will be noticed that, in this case, the common chord P Q is equidistant from the directrices. For the distance of P from each directrix is equal to SP . THE PARABOLA. 31 43. Prop. XV. The circle passing through the points of intersection of three tangents passes also through the focus. Let Q, P , Q be the three points of contact, and F , T , F the intersections of the tangents. In Art. (36) it has been shewn that, if F P , F Q be tangents, the angle SQF = SF P.