. The principles of projective geometry applied to the straight line and conic . 46 Principles of Projective Geometry 30. The important theorem of Art. 25—if for the present the use of imaginarypoints be excluded—requires the two circles to intersect in a pair of real the following theorem it may be ascertained when this will be the case. Ay A, B, B are four colliiiear points and circles are described on A A and BBto contain angles a and /3 respectively. (1) If the points occur in the order ABAB, the circles will always intersect inreal points. (2) If the points occur in the order
. The principles of projective geometry applied to the straight line and conic . 46 Principles of Projective Geometry 30. The important theorem of Art. 25—if for the present the use of imaginarypoints be excluded—requires the two circles to intersect in a pair of real the following theorem it may be ascertained when this will be the case. Ay A, B, B are four colliiiear points and circles are described on A A and BBto contain angles a and /3 respectively. (1) If the points occur in the order ABAB, the circles will always intersect inreal points. (2) If the points occur in the order AABB, the circles will intersect in realpoints, if {BBAA)> (3) If the points occur inpoints, if 1 + /3-cua ~— order ABBA, the circles will intersect in real {BBAA)<- oa + ^ (1) In this case the theorem is (2) Let G and D be the centres of the circles, M and iV the feet of theperpendiculars for these points on AABB, and S a point of intersection of thecircles. Then the condition that S should be real is that {CS+SDf>CD^ (i). Then angle J Cif= angle MCA = a, and angle BDJV=SLng\e NDB = (i. Projection and Perspective 47 Let 0 be any point on the line AABB and CAthe pei-pendicular from C on (i) becomes {CS+SDf>I)K^+CK- (ii). But CS=^4^^, ,Si)=^^^, 2 sni a 2 sin /S CK=^ + 05 _ OA + OA ^ OB-OB _ OA-OA 2 2 2tan/3 2tana Hence (ii) becomes /OA-OA OB- OBV fOB-OB _ OA-OA\^V 2 sin a 2 sin /3 / ^ V 2 tan ii 2 tan a / {OA-OA) {OB-OB) (OB + OB _ o^ + Qjy 1 + cos a cos j8 + sin a sin/3 sin a sin fi >2{0B-0A){0B-0A); l+cos(a-^) ^ ^rniiD\■■ cos(a-^)-cos(a + ^)^CT7^ ^ > (^^^^^^)- Hence the required condition is ,a-/3cos^ -- {BBAA)> cos^-—— (3) As in (2) it may be shown that the condition in this case is that i2_ ^BAAB)> ---^^^^l-J---> COS {a-^)-COS {a + (i) .
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