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By Robert Lachlan

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

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Show same plane, «how straight line, and four straight lines in the = (abc) + (cda). also that (abc) = (dbc) + (dca) + (dab). // A, B, C be any three points on a P, Q any other points in the same plane with 35. AB = 0. Let x denote the straight But, by line PQ. AB = 0. PQ. §21, 2—2 : ELEMENTARY THEOREMS 20 Therefore Ax Bx Cx = (APQ) (BPQ) (CPQ). AB = : Hence : : : . (1). AB This relation Ex. 36. C be any three points on a straight any given point ; show that If J, B, 1. straight line, and (2), (3).

AE; this relation if . : . 1 . : : hold, the points D, E F } must be collinear. Ex. Show 7. Points that the Ex. 8. P and Q are taken on two straight lines AP :PB=CQ:QD. sum The of the areas (PCD), sides points D, E, F. BC, CA, Show point, A, B, C, If 9. is that constant. AB of a triangle meet any straight line in the P can be found in the line DEF such th*t that a point the areas (PAD), (PBE), Ex. (QAB) AB, CD, such D (PCF) [St John's Coll. ] are equal. be any four points on a and circle P be any given show that PA* (BCD) - PB2 (CDA )+PC2 (DA B)-PDi .

When 41. P the segment AB is divided harmonically in the points Q, the range {AB, PQ] is called a harmonic range and the pairs of points A, B, P, Q; are called conjugate points of the range. and HARMONIC RANGE. -2r, {AB, PQ\ for a being inserted to distinguish the pairs It will be found convenient to use the notation comma harmonic range, the of conjugate points. 42. Ex. ABC If 1. BAC divide the base be any triangle, show that the bisectors of the angle BC harmonically. Ex. 2. If tangents OP, OQ be drawn to a circle from any point 0, and if cut the circle in the points 11 any straight line drawn through the point and S and the chord PQ in the point V, show that {0 V, RS} is a harmonic range.

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An Elementary Treatise on Modern Pure Geometry by Robert Lachlan


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