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ABCD algebraic altitude anharmonic ratio axis base bisect called centre chord circle circumference common point complete constant construct demonstration describe determine diagonals diameter difference distance divided draw equal equations equivalent extremities faces figure formed four geometrical give given circle given line given point greater harmonic Hence hypotenuse included inscribed intersection isoperimetric isosceles let fall line drawn line joining line passing locus manner maximum mean meet middle point opposite sides parallel passing pencil perpendicular plane polar pole polygon position preceding problem produced proof proposition Prove quadrilateral radical radius rectangle regular relation respect right angled right angled triangle right line sides similar solid solution square straight line student SUG's symmetrical Take tangent third transversal triangle vertices whence
Page 292 - Find the locus of a point such that the sum of the squares of its distances from two fixed points shall be equivalent to the square of the distance between the fixed points.
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 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 270 - To describe a circle with a given radius, which shall pass through a given point and be tangent to a given line.
Page 249 - From this proposition it is evident, that the square described on the difference of two lines is equivalent to the sum of the squares described on the lines respectively, minus twice the rectangle contained by the lines.
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 294 - O'O" as at O. With 0 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.] SYS.
Page 267 - The intersection of two spherical surfaces is the circumference of a circle whose plane is perpendicular to the line joining the centres of the surfaces and whose centre is in that line. Let 0, 0' be the centres of the spherical surfaces, and let a plane passing through 0, 0' cut the spheres in great circles whose circumferences intersect in the points A and B.
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...