# An Elementary Treatise on Modern Pure Geometry by Robert Lachlan

By Robert Lachlan

This quantity is made from electronic photographs from the Cornell collage Library historic arithmetic Monographs assortment.

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Ox + DO. Ax+OA. Dx = 0. DO. Ax-CO AD. Dx=0 DO BC Ax+DA CO Bx- BO DA Cx-BC AO Dx=0. (BCD)=\$D0 BC. sin BOD, (CDA)=\$DA CO sin AOC, (DAB)=\DA BO sin AOC, (ABC) =\BC A sin BOD. sin BOD = - sin A OC. . But line, . . . . ; . . Also Hence we have the required 37. If A, B, C, D be a straight . . result. any four points in a plane, the locus of a sum of the areas (PAB), (PCD) is point P, which moves so that the constant, is . line. CONCERNING AREAS. and 23 Let the straight lines AB, CD meet in the point 0, an

39 . OA'. also proceed otherwise. Let any two circles In- drawn passing through the points A, A', and the points B, Bf respectively; and let these circles intersect in the points X and 1". Then if the line meet the given straight line in the point 0, XY this point will be the centre of the range. For evidently OA . OA' = OX 0Y= OB To obtain the conjugate point draw the will cut 70. OC Ex. 1. between segments A A A', OB'. any point C, we have merely This to circle C, the required point. For lies to .

AND PENCILS. 17 The six angles of any pencil of four rays 30. \ABCD) art connected by the relation sin BOC . sin A OD + sin CO A . sin 110 1> + sin AOB. sin COD -0. O Let any straight in line the points A, B, C, D. be drawn cutting the rays of the pencil Then, by § 25, we have BC. AD + CA But if ON on the line . BD + AB CD = 0. AB = 0A. OB. sinAOB, for NO AD, NO CD, &c. . , above relation, we obtain the relation BOC . sin A OD + sin CO A This relation is . of great use. sin BOD + sin A OB .