A Treatise on Special Or Elementary Geometry: An advanced course in geometry

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Sheldon, 1872 - Geometry - 84 pages
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Page 294 - O'O" as at O. With O as a centre and OA as a radius, describe a circle. Then is this circumference the locus required. For, let BC be any secant line passing through A, we may show that P is the middle point of BC- [Having done this, as above, and shown that any point not In this circumference is not the middle of the secant line passing through A, his solution is complete.] 875.
Page 292 - Show that the locus of a point such that the sum of the squares of its distances from two fixed points is constant, is a circle.
Page 245 - If from any point there extend two lines tangent to a circumference, the angle contained by the tangents is double the angle contained by the line joining the points of contact and the radius extending to one of them.
Page 310 - Three lines are in harmonical proportion, when the first is to the third, as the difference between the first and second, is to the difference between the second and third ; and the second is called a harmonic mean between the first and third. The expression 'harmonical proportion...
Page 250 - The square described on the hypothenuse of a right-angled triangle is equivalent to the sum of the squares described on the other two sides.
Page 246 - Prove that the sum of the angles of a regular five point star (Fig. 101) is two right angles. Show, also, that the figure formed by the intercepted portions of the lines is a regular pentagon. 638. If the sides of a regular hexagon are produced till they meet, show that the exterior figures will be equilateral triangles.
Page 272 - If from the vertical angle of a triangle three straight lines be drawn, one bisecting the angle, another bisecting the base, and the third perpendicular to the base, the first is always intermediate in magnitude and position to the other two.
Page 275 - A' parallel to A, FIG. 427. and the figure will suggest the construction. 803. To pass a plane through a given line and tangent to a given sphere. SUo's.—Pass a plane through the centre of the sphere and perpendicular to the given line. Through the point of intersection and in this secant plane draw tangents to the great circle in which the secant plane intersects the surface of the sphere. The points of tangency will be the points of tangency of the required planes (?), of which there are thus...
Page 252 - The sum of the squares of the four sides of any quadrilateral is equal to the sum of the squares of the diagonals plus four times the square of the line joining the mid-points of the diagonals.
Page 246 - ... 641. The side of an equilateral triangle inscribed in a circle is equal to the diagonal of a rhombus, whose other diagonal and each of whose sides are equal to the radius.

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