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Exercises on Euclid and in Modern Geometry for the Use of Schools, Private Students, and Junior University Students

Exercises on Euclid and in Modern Geometry for the Use of Schools, Private Students, and Junior University Students


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This historic book may have numerous typos and missing text. Purchasers can download a free scanned copy of the original book (without typos) from the publisher. Not indexed. Not illustrated. 1881 Excerpt: ...from P or Q. N.B. If the circle about the triangle APG (Figs. 1 and 2) do not pass through C let it cut CQ in S. Therefore PQ.QG = AQ.QS, but PQ.QG = AQ. QC (constr.), therefore QS = QC, which is impossible; therefore if PQ. QG = AQ. QC, the four points A, P, G, C must lie in the same circumference. 144. In a given circle inscribe a triangle, having its base parallel to a given line and its two sides passing through two given points, not both situated on a line parallel to the given line. Let P, Q be the two given points, and PM the line to which the base is required to be parallel. At any point V in the circumference of the given circle, make A the angle R VS equal to the given angle GPM, then the chord ES is constant. Make PQ. QG equal to the square on the tangent from Q to the given circle, and through G draw GH and GF such that HC and FK shall be each equal to B.S (58). Join QC and QF, and let these lines (produced, if necessary) meet the circle again in A and D. Join AP, PD, meeting the circle again in B and E. Join BC and FE. ABC and DEF are two triangles inscribed as required. For join BR and EK, and produce BC and PQ to meet in L. Because PQ.QG = AQ. QC (constr.), therefore the quadrilateral APGC is circumscribable by a circle, and therefore (III. 22) the angles CGQ and BAC are equal, but BAC and BHC in the same segment are equal. Therefore the angles CGQ and BHC are equal, and therefore (I. 27) BH and PQ are parallel. Therefore the angles HBC (or R VS) and CLP are equal, but HBC and GPM are equal. Therefore CLP and GPM are equal, and therefore BC is parallel to PM. Hence, ABC is a triangle inscribed as required. Again, because PQ. QG = FQ. QD (constr.), therefore the four points F, D, P, G lie in the same circumference, therefore the angles FGP and...


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Product Details
  • ISBN-13: 9781130530780
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 82
  • Spine Width: 4 mm
  • Width: 189 mm
  • ISBN-10: 1130530787
  • Publisher Date: 01 Mar 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 163 gr


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