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angles angular distances angular radius anharmonic ratio Apply arcs joining axes base bisectors called centre centre of similitude chord circle joining circle passing circumcircle colunar common tangents concurrent concyclic conjugate connected cosc cosines curve denote described determine diagonals direction divide drawn equal equation Examples expression external figure fixed point follows four points given given circles gives harmonic Hence imaginary inscribed internal intersection inverse involution locus meet middle points obtained opposite sides origin orthogonally pair pass pencil perpendicular plane plano points of contact polar polar circle pole polygon position prove quadrant quadrilateral radical axis radii rays reciprocal relation respect segments sides Similarly sines small circle sphere spherical triangle tangents theorem third triangle ABC variable vertex vertices
43. lappuse - EXCESS (Lat. excessus, departure, from excedere, to depart, from ex, out + cedere, to go). The remainder arising from dividing one number by another is often called the excess, as in casting out nines in the test for divisibility. (See CHECKING.) In spherical trigonometry the excess of the sum of the angles of a spherical polygon over n 2 straight angles (the sum of the angles of a plane polygon of the same number of sides) is called the spherical excess of the polygon; eg, the spherical excess...
55. lappuse - ... respectively perpendicular to GC' and GC" : then construct a triangle with the lines HD, DC', and HC" ; the angle HCD will be equal to the angle C of the spherical triangle. It is plain that the angle EBE' is equal to the angle B of the spherical triangle, and the angle EAE
45. lappuse - Trip. 1928) 30. Show that, if the sum of the lengths of the hypotenuse and another side of a right-angled triangle is given, then the area of the triangle is a maximum when the angle between those sides is 60°. (Math. Trip. 1909) 31. A line is drawn through a fixed point (a, 6) to meet the axes OX, OY in P and Q.
113. lappuse - A line cutting a circle and passing through a fixed point is cut harmonically by the circle, the point, and the polar of the point.
5. lappuse - Questions and exercises on Chapter IV. will be found on page 105. CHAPTER V. CIRCLES CONNECTED WITH SPHERICAL TRIANGLES. 49. The circumscribing circle. The circle passing through the vertices of a spherical triangle is called the circumscribing circle, or circum-circle, of the triangle. This circle can be constructed in somewhat the same manner as the circumscribing circle of a plane triangle. Let ABC (Fig. 41) be a spherical triangle, and let S denote the radius (ie the polar distance, Art.
29. lappuse - The volume of this cone is found by multiplying the area of its base by one-third of its height ; the height being equal to the radius of the sphere.
111. lappuse - B be two points, such that the polar of A passes through B, then the polar of B passes through A.
90. lappuse - When three triangles are two by two in perspective, and have the same axis of perspective, their three centres of perspective are collinear. Let abe, a'b'c', a"i"c"be the three As whose corresponding sides are concurrent in the collinear points A, B, C.
163. lappuse - If the three sides of a plane triangle be replaced by three circles, then the circles touching these, which correspond to the inscribed and escribed circles of a plane triangle, are all touched by another circle.