A first course in projective geometry . N, so the subtangent NT is t\vice the abscissa AN ^ \ p^ Y \ w /A /^ f ^ G Fig. 90. (2) SG = SP(§2, Cor. 1)=PK = NX. /. NG = SX = 2AS. (3) Y and Y, the feet of the perpendiculars from S on UP,UP, lie on the tangent at the vertex (§ 3, Cor. 1). .*. SYYU is a cyclic quadrilateral, and angle SYY = angle SU YBut angle SYY= angle STY = angle TPK = angle SPT. .*. angle SPU = angle , by § 1, angle USP = angle USP. .*. the As SUP, SUP are equiangular and similar. TANGENT AND NORMAL PROPERTIES 173 (4) If UVW be a triangle formed by tangents, its circum-ci
A first course in projective geometry . N, so the subtangent NT is t\vice the abscissa AN ^ \ p^ Y \ w /A /^ f ^ G Fig. 90. (2) SG = SP(§2, Cor. 1)=PK = NX. /. NG = SX = 2AS. (3) Y and Y, the feet of the perpendiculars from S on UP,UP, lie on the tangent at the vertex (§ 3, Cor. 1). .*. SYYU is a cyclic quadrilateral, and angle SYY = angle SU YBut angle SYY= angle STY = angle TPK = angle SPT. .*. angle SPU = angle , by § 1, angle USP = angle USP. .*. the As SUP, SUP are equiangular and similar. TANGENT AND NORMAL PROPERTIES 173 (4) If UVW be a triangle formed by tangents, its circum-circle will pass through S. For, considering the tangents fromV, we have by the preceding, angle SVW = angle SPV. But SUW was proved equal to this angle. .*. SUVW is a cyclic quadrilateral. § 9. The Director Circle. The locus of a point from which orthogonal tangents canbe drawn to a conic is a concentric circle. Let TYY, TZZ (Figs. 91a and h) be orthogonal tangentsfrom T to a central conic. The feet of the perpendiculars from T y; ^. Pig. 91a. the foci, viz. Y, Y, Z, Z, lie on the auxiliary circle. Moreover, SYTZ, SYTZ are rectangles, and TY . TY = SZ . SZ = BC^. Also TY. TY = square of tangent from T to the auxiliary circle = CT^ - sq. of radius of aux. 0 (for the ellipse), but = sq. of radius of aux. O - CT^ (for the hyperbola) (for in the latter case the foci lie on opposite sides of any 174 PROJECTIVE GEOMETRY tangent, and it follows that T necessarily lies between Yand Y, Z and Z). .-. CT- = AC- 4- BC- (for the ellipse)and AC^ - BC- (for the hyperbola). .*. CT is constant in both cases, and the locus of T is acircle. This is called the Director Circle.
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