Mathematical Questions with Their Solutions, from the "Educational Times" Volume 12

Mathematical Questions with Their Solutions, from the "Educational Times" Volume 12

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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. 1901 edition. Excerpt: ...A, 0 This determinant is, when worked out, (/+ g cos C) (g + h cos A) (A +/cos B) + (A + g cos A) (/+ h cos B) (g +/cos C), which breaks up identically into the product of sin A + g sin B + h sin C into gh sin A + A/sin B + _/j sin C; this result determining the locus of (/, g. A), according to what we all know, to be the circle circumscribing the original triangle. The geographical statement of this theorem homographically transformed, runs thus: Consider two fixed points joined by a line intersecting the three sides of a triangle; to each point of intersection successively let there be taken the harmonic conjugate with the fixed points as a pair of conjugates: join each point so determined with any point on the conic passing through the fixed points and the vertices of the triangle; the intersections of these connectors each with the "corresponding" side of the original triangle, lie in a straight line. And, analytically stated, the proposition is as follows; Let (a, /8, 7), (a', /8', 7') be the two fixed points, and (/, g, h) the arbitrary point on the conic circumscribing the triangle of reference. The coordinates of the "harmonic conjugates" spoken of above are as follows; viz., x: y: z = y'a. + 7o' /$ + yff: 277' for that corresponding to the point of intersection on the side x, and similar expressions for the other two. The coordinates of the point where the line joining this conjugate with the point (f, g, h) intersects the side 2, are x: y: z = h (7'a + 7o')--2/77' h (y'0 + y0')--2gyy' 0, with similar expressions for the other two points of intersection; and the condition that these three should lie in a straight line is 0, f(a'0 + a$')-2gaa', f (a'y + ay')-2haa' 1 0 03a' + iS'o)-ye/', 0, g (#7 + 0Y)-2 more

Product details

  • Paperback | 36 pages
  • 189 x 246 x 2mm | 82g
  • English
  • Illustrations, black and white
  • 1236849132
  • 9781236849137