Annals of Mathematics Volume 1-2

Annals of Mathematics Volume 1-2

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This historic book may have numerous typos and missing text. Purchasers can usually download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1884 edition. Excerpt: ...1. Caustructiou of the triangle This part of the problem is old. 2. Calculation of the area. Let /V/, r be the given perpendicular, bisector, and median, and P, Q, R their points of termination. Put ljuji for OR, RP, PQ respectively and 2x for the base. Then the area is px and x is determined from the condition Euc. VI, 3 x-t I = 1 p--r (' + mf x-l p-+ x--;//)2' lr--////"' whence IV. M. Thornton 29 Given a pair of points A.B, if C%D are such that OD. 0C= OA2 = OB2 and AOC = ADD, prove that the pair AyB bears similar relation to the pair C, D. Show, also, the existence of a pair E, F which bears the same relation to each of the pairs AyB and C, D. Wm. Woolscy Johnson SOLUTION. I. Assume 0 to be the middle point of AB. As an easy deduction from the conditions given, the triangles CO A and AOD are similar; also the triangles COB and BOD. Z CAD = Z CAO + Z OCA = Z COB = Z BOD and Z CBD = Z CBO + Z OCR = Z COA = Z AOD also OB = )AO: CB:: OD: BD and (04 = ) BO::: OD:,4;. '. the triangles AOD and CBD are similar; also the triangles BOD and CAD are similar. If P be a point in CD such that Z ZMZ = Z C, since Z C#Z = Z A CD, the triangles ZMZ and CAP art similar. Dividing OB( = AO): OD:: AC: AD:: CB: BD by AP: AD:: CP: BD, we have 2C: AP:: CB: CP; and, since Z OAD = Z C4/-= Z 5 CD, the triangles and /JCP are similar, and: CP:: CP: BP, or AP / BP = CP'1; similarly it may be shown that AP X BP--DP1. 2. Let CA be a line parallel to through A, and 0 a point such that Z 0C = Z CMC and BQ: CQ:: BD: CA; then the triangles iZ ? and C/42 wiU De similar and also the triangles CQB and AQD. Let /m2 be a line bisecting Z g/? and put /m2 = AQ X /2; continue, F2 to the point such that QE =;then will E and F satisfy the conditions required. more

Product details

  • Paperback | 70 pages
  • 189 x 246 x 4mm | 141g
  • United States
  • English
  • black & white illustrations
  • 1236879392
  • 9781236879394