If two circles cut at right angles shew that the intercept made by either circle on any line drawn through the centre of the other circle is divided internally and externally in the same ratio by the points in which it meets the other circle. If the two circles intersect in \(A\) and \(B\) and the diameters of the circles through \(A\) meet them again in \(X\) and \(Y\) shew that the circle through the mid-points of the sides of the triangle \(AXY\) passes through \(A\) and \(B\).
Shew that the four circles which circumscribe the triangles formed by three out of four given lines meet in a point. Hence shew that the five spheres which circumscribe the tetrahedra formed by four out of five given planes meet by fours in five points, one in each plane.
Forces \(P_1, P_2, P_3, P_4, P_5, P_6\) act along the sides of a regular hexagon taken in order. Shew that they will be in equilibrium if \(\Sigma P = 0\) and \(P_1-P_4 = P_3-P_6=P_5-P_2\).
Prove that, if \(ax^2+2bx+c\) is to be positive for all real values of \(x\), it is both necessary and sufficient that
Given a straight line \(AB\) divided into two segments by a point \(P\) shew that the locus of points at which these two segments subtend equal angles is a circle. If a line through \(B\) meets this circle in \(C\) and \(D\) shew that \(CA\) and \(DA\) are equally inclined to \(AB\).
\(A, B, C, D\) are four points in a plane, and \(A', B', C', D'\) are the circumcentres of the triangles \(BCD, ACD, ABD, ABC\) respectively. Prove that a conic having \(D\) and \(D'\) as foci will touch \(B'C', C'A', A'B'\).
\(AB, BC\) are two uniform heavy rods of equal length and weight \(W\). The rod \(AB\) can move freely about \(A\), there is a hinge at \(B\), and at \(C\) there is a ring which can move freely along a fixed rod through \(A\) which is inclined downwards at an angle \(\alpha\) to the horizontal. Shew that in equilibrium \(\tan BAC = \frac{1}{2}\cot\alpha\), and that the horizontal component of the action at \(B\) is \(\frac{1}{2} W \sin 2\alpha\).
Give an account of the method of Inversion, as applied to plane geometry, shewing its effect upon straight lines, circles and systems of coaxal circles, and proving that magnitudes of angles are unchanged, and that pairs of points inverse with respect to any circle retain the property after inversion. The centre of inversion being the origin, find the equation of the curve inverse to the ellipse \[ 2x = ax^2+by^2+2hxy \] and shew that the asymptote is the inverse of the circle of curvature at the origin. State the property of circles inverse to (i) the circles of curvature of the ellipse, (ii) the circles which touch the ellipse at two points, considering in particular those circles of the families (i) and (ii) which pass through the origin.
Express \[ \frac{57x^3 - 25x^2 + 9x - 1}{(x-1)^2(2x-1)(5x-1)} \] as a sum of partial fractions; and expand in ascending powers of \(x\) as far as the term in \(x^4\).
(i) \(AOA', BOB'\) are two chords of a conic, and \(P, Q\) are two points on a line through \(O\). Shew that, if \(AP\) and \(BQ\) meet on the conic, \(B'P\) and \(A'Q\) will do the same. (ii) \(A, B, C\) are three points on a given conic and \(O\) is a point on a given line. \(AO, BO, CO\) meet the conic again in \(A', B', C'\), and \(BC, CA, AB\) meet the line in \(A'', B'', C''\) respectively. Shew that the lines \(A'A'', B'B'', C'C''\) meet in a point that lies on the conic, and that, if any conic is drawn through \(A, B, C, O\), its two remaining intersections with the line and the conic are collinear with this point.