By Bibhutibhushan Datta
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Extra info for Ancient Hindu Geometry: The Science Of The Sulba
B'C CA' + A'B'. BC CA =0, BC CA' + A'B B'C CA = 0. AB'. . . Conversely, if any one of these relations hold, then the 75. range [A A', BB', CC) be in involution. will For if not, let a range in involution be formed so that A, A', and B, B', are conjugate couples and let C" be the point conjugate ; to the point C. Then if we have given the AB' we shall also C B'C CA, . CA' = - A'B. B'C. CA. BC:CA= BC" C"A : must coincide with C [AA\ is . CA' = - A'B . ; that BB', the range is, CC] in involution.
Ex. 1. straight line, must hold may If A, B, C, must There- in all cases. be stated in the forms also it : BC = AC-AB (2). BC=BA+AC (3). +HK+KA=0. Ex. 2. If A, B, C be any three points on the be the middle point of BC, show that AB + AC=2A0. same straight line, and if B : METRICAL PROPERTIES OF RANGES. Ex. D If A, B, C, 3. be points on the same CD AB, middle points of respectively, line, 15 and if ,\', }' Um 1* show that 2XV=AC+ BD = AD + BC. Ex. C A, B, be points on the same line, and if A', B, C 1* respectively the middle points of the segments BC, CA, AB, show that If 4.
In the same way, if OQ we can show cut O'P' in p', then p' , b, q' that if will OP cut O'Q' in q', and be collinear. D Ex. 1. Show that if A, B, C, 63. be any four points in a plane, and if the six lines joining these points meet in the points E, F, G then the two ; POINTS CONJUGATE TO TWO POINT-PAIRS. 86 any one of these points are harmonically conjugate with EFG which meet in the same point This follows from § 60. lines which meet in the two sides of the triangle Ex. Deduce from 2. corresponding theorem when four straight § 62, the lines are given.
Ancient Hindu Geometry: The Science Of The Sulba by Bibhutibhushan Datta