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Geometrical Exercises for Beginners

Geometrical Exercises for Beginners


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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. 1882 Excerpt: ...first equation by m, and the second equation by n, and adding, we have mAC2 + nBC2 = mAO2 + nBO2 + (- + -) CO2 + 2OD(mAO-nBO). But mAO obviously equals nBO, since BO contains m, and AO n equal parts; therefore the last term of the equation, viz. 2OD (mAO--nBO), vanishes. Hence mAC2 + nBC = mAO2 + nBO2 + (m + n) 002. Example. Given, in magnitude and position, the base of a triangle, and that m times the square of one side together with n times the square of the other is equal to a given square; find the locus of the vertex. Divide the base, AB, into m + n equal parts (18); let AO contain n parts, BO m parts; then AO, BO, and the point 0 are fixed. Hence, in the above equation, the only unknown quantity is OC, which can immediately be determined. The locus is, then, a circle, with 0 as centre, and a radius OC of known length. Prop. 100. The area of a triangle is equal to Js (s--a) (s--b) (s--c), where s is equal to half the perimeter. Let (see fig. Prop. 23) O be the centre of the inscribed circle, and 0, the centre of the escribed circle opposite the angle A, of the triangle ABC. Draw Of, OxF perpendiculars to AB. Since (5) the angle OBOX is right, the sum of the angles OBf, OtBF is equal to a right angle; but the sum of the angles OBf, BOf is also a right angle; therefore the angles OJiF, BOf are equal; and the angles BfO, BFOx are equal, each being right; therefore the triangles BOf, BOjF are equiangular. Hence (VI. 4) Of-.Bf-.-.BF-.O.F, or r: s--b:: s--c: rX (see Prop. 23, Exs. 1, 2), that is, rrX = s--b) (s--c). But sr = (s--a) rx = A, the area of the triangle (see Prop. 23, Exs. 4, 5); therefore 8. (s--a) rrx = A'; but rrx =(s--b)(s--c), therefore s.(s--a) (s--b) (s--c) = Aa, 4. Prove that, of an equilateral triangle, the radius of an escribed c...



Product Details
  • ISBN-13: 9781130751659
  • Publisher: Rarebooksclub.com
  • Publisher Imprint: Rarebooksclub.com
  • Height: 246 mm
  • No of Pages: 46
  • Spine Width: 3 mm
  • Width: 189 mm
  • ISBN-10: 1130751651
  • Publisher Date: 01 Mar 2012
  • Binding: Paperback
  • Language: English
  • Returnable: N
  • Weight: 100 gr

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