Prove the Simson's Line theorem for a triangle inscribed in a circle, namely that the feet of perpendiculars to the sides of the triangle from a point \(P\) on the circumference of the circumcircle are collinear. Prove also that the Simson Line of \(P\) bisects the join of \(P\) to the orthocentre of the triangle.
The sides \(BC, CA, AB\) of a given triangle \(ABC\) are cut by a straight line \(l\) in points \(A'\), \(B'\), \(C'\) respectively, and \(BB', CC'\) meet in \(A''\), \(CC', AA'\) in \(B''\), \(AA', BB'\) in \(C''\). Prove that \(AA'', BB'', CC''\) meet in a point \(L\). Show that for a variable line \(l\) through a fixed point \(O\), the locus of \(L\) is a conic through \(A, B, C\).
Define coaxial circles, and from the definition prove that any circle orthogonal to two circles of one system of coaxial circles belongs to a second system of such circles. Prove that no two circles orthogonal to two other circles with real points of intersection can themselves have real points of intersection. Show how to construct the radical axis of two given circles, distinguishing cases with four, two, or no real common tangents.
Points on the surface of a sphere are projected from a vertex \(O\) of the surface onto a plane through the centre of the sphere and perpendicular to the radius to \(O\). Prove that in general circles on the sphere project into circles on the plane, mentioning any particular exceptions: and that the angle of intersection of two curves on the sphere is equal or supplementary to the angle between the corresponding curves on the plane. State the nature of the curve on the plane into which will project a curve on the sphere cutting the great circles through \(O\) at a constant angle.
Prove Brianchon's Theorem, that the joins of opposite vertices of a hexagon circumscribed about a conic are concurrent. Hence, or otherwise, prove that if \(l_1, l_2, l_3, l_4\) are four tangents to a parabola, then the straight line parallel to \(l_4\) through the intersection of \(l_1\) and \(l_2\) and the straight line parallel to \(l_1\) through the intersection of \(l_3\) and \(l_4\) intersect on the diameter through the intersection of \(l_2, l_3\).
Using line (tangential) coordinates \(l, m\), interpret the following line equations:
Prove that if two triangles \(ABC\) and \(A'B'C'\) are in perspective from \(O\), the intersections of corresponding pairs of sides are collinear in an axis of perspective corresponding to \(O\). If, with respect to a conic \(S\), \(B'C'\) is the polar of \(A\), \(C'A'\) the polar of \(B\), and \(A'B'\) the polar of \(C\), prove that the triangles are in perspective, and that the vertex and axis of perspective are pole and polar with respect to the same conic.
Prove that if two pairs of opposite vertices of a quadrilateral are conjugate with respect to a conic, then the third pair has the same property. Prove that in general for a given quadrilateral there is one and only one conic to which the diagonal triangle of the quadrilateral is self polar and such that the pairs of opposite vertices are conjugate.
(i) Solve the equation \[ \tan\theta + \sec2\theta = 1. \] (ii) Sum the infinite series \[ 1 - \frac{1}{2!}\cos2\theta + \frac{1}{4!}\cos4\theta - \dots. \]
Prove that if the sides of a plane quadrilateral are of given lengths \(a, b, c, d\), then the area enclosed is greatest when the quadrilateral is cyclic, and its value is then given by \(\sqrt{(s-a)(s-b)(s-c)(s-d)}\), where \(2s=a+b+c+d\).