Prove that the Simson line of a point D on the circumcircle of a triangle ABC bisects the join of D to the orthocentre of the triangle. Hence, or otherwise, show that the four Simson lines, obtained by taking each of A, B, C, and D in turn with the triangle formed by the other three points, are concurrent.
Prove that the inverses of two orthogonally intersecting curves are orthogonal. Show that the inverse of a circle with respect to any point as centre of inversion is either a circle or a straight line. Explain what figure is obtained by inverting a set of non-intersecting coaxal circles with respect to one of their limiting points, and show that any circle cutting all circles of the system orthogonally must pass through both limiting points.
If \(a, b, c,\) and \(d\) are any four coplanar straight lines in general position, and if O is the second point of intersection of the circumcircles of the triangles formed by the triads \(abc\) and \(abd\) respectively, prove that the circumcircles of the triangles \(acd\) and \(bcd\) also pass through O. Illustrate this result by the case of four tangents to a parabola, determining in particular the identity of the point O.
Prove that if in the tetrahedron ABCD, AB=CD and AD=BC, then AC and BD are bisected by their mutual perpendicular. Prove for a general tetrahedron that the joins of midpoints of opposite edges are concurrent, and show that the join of the midpoints of one pair of opposite edges can only be their mutual perpendicular if both the other pairs of opposite edges are equal in length, as above.
Prove that if two rectangular hyperbolas can be drawn through four points, then every conic through these points is a rectangular hyperbola. Hence, or otherwise, show that the orthocentre H of a triangle inscribed in a rectangular hyperbola lies on the curve. Show further, that if D is the fourth point in which the circumcircle of the triangle meets the curve, then DH passes through the centre of the rectangular hyperbola.
Prove that the ratio of the intercepts PG, PH on a normal at a point P to a central conic between P and the points G and H where the normal cuts the two principal axes is constant. Prove that for a point on any one of a system of confocal conics the product of the intercept on the normal between the two principal axes and the length of the projection on the normal of the radius from the centre of the conic to the point is constant.
Prove that pairs of tangent rays drawn from a point L to conics touching the sides of a quadrilateral are in involution, and that the joins of L to opposite pairs of vertices of the quadrilateral are members of the same involution. Hence, or otherwise, prove that if two pairs of opposite vertices of a quadrilateral are conjugate with respect to a conic S, the third pair of opposite vertices is also conjugate with respect to S.
Prove that in general two conics have one and only one common self conjugate triangle. Prove that if two quadrilaterals have the same diagonal triangle, the eight sides of the two quadrilaterals touch a conic.
The two diagonals AC and BD of a plane quadrilateral meet in O. Prove that \[ \text{area } \triangle\text{AOB} \times \text{area quadrilateral ABCD} = \text{area } \triangle\text{ABC} \times \text{area } \triangle\text{ABD}. \]
(i) Prove that if \(n\) is an odd integer, \(\sin n\theta + \cos n\theta\) regarded as a rational integral function of \(\sin\theta\) and \(\cos\theta\) is divisible either by \(\sin\theta+\cos\theta\), or by \(\sin\theta-\cos\theta\). (ii) Prove that if \(m\) and \(n\) are two different odd integers, or two different even integers, \(m\sin n\theta - n\sin m\theta\) is divisible by \(\sin^3\theta\).