D, E, F are the mid-points of the sides BC, CA, AB of a triangle, Y and Z are the feet of the perpendiculars from B and C on to the opposite sides, and YZ meets FD in M and DE in N. Prove that EF is a common tangent of the circles EYN, FZM and that AD is the radical axis of the circles.
A closed convex curve C lies entirely inside a convex polygon P. Prove that the perimeter of C is less than that of P.
Prove that, if a finite set of points in space possesses an axis or a plane of symmetry, then the centroid of the points (i.e. the centre of mass for equal masses placed at the points) lies on the axis or in the plane. Deduce that if the points are such that, for each pair PQ of them, there exists an axis or plane of symmetry of the set with respect to which P and Q are images, then all the points lie on the surface of a sphere.
Show that there is just one point P in the plane of the parabola \(y^2=4ax\) such that the three normals from P to the parabola make angles of \(60^\circ\) with each other, and find the coordinates of P.
If the eccentric angles of the four points of intersection of an ellipse and a circle are \(\alpha, \beta, \gamma, \delta\), prove that \(\alpha+\beta+\gamma+\delta\) is a multiple of \(2\pi\). Show that three circles can be drawn through a point P of an ellipse to osculate the ellipse elsewhere, and that the circle drawn through the three points of osculation contains P.
Two conjugate diameters of a conic meet the polar of a point P in Q and Q', and the perpendiculars to the diameters at Q, Q' meet in P'. Prove that P and P' are conjugate points with respect to the director circle of the conic.
A parabola is drawn to have four-point contact with a central conic S at P. Prove that the diameter of S through P is parallel to the axis of the parabola; and that if S is a circle P is the vertex of the parabola.
Three concurrent lines OA, OB, OC are cut by a transversal ABC. P and Q are two points on OA; PB meets OC in R, QC meets OB in S, PC meets OB in X, and QB meets OC in Y. Prove that RS and XY meet on OA.
Prove that if ABC is a triangle inscribed in a conic S, then the tangents to S at A, B, C meet the opposite sides of the triangle in three collinear points. Show that a conic can be projected into a circle in such a way that the projection of a given inscribed triangle is equilateral.
Show that any tangent to one of the three conics \[ x^2+2yz=0, \quad y^2+2zx=0, \quad z^2+2xy=0 \] is divided harmonically at its intersections with the other two. Prove, further, that the polar reciprocal of any one of these conics with respect to one of the others is the third conic of the set.