Find the point of intersection of the tangents to the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] at the points whose eccentric angles are \(\alpha\) and \(\beta\). Show that the area of the triangle formed by the tangents at the points whose eccentric angles are \(\alpha, \beta, \gamma\) respectively is \[ ab \tan\tfrac{1}{2}(\beta-\gamma) \cdot \tan\tfrac{1}{2}(\gamma-\alpha) \cdot \tan\tfrac{1}{2}(\alpha-\beta). \]
\(P, Q, R\) are points on a rectangular hyperbola. Prove that the centre of the hyperbola lies on the nine-points circle of the triangle \(PQR\). A circle and a rectangular hyperbola intersect in four points, and one of their common chords is a diameter of the hyperbola; show that another of them is a diameter of the circle.
Prove that, in a right-angled triangle, the square described on the hypotenuse is equal to the sum of the squares described on the other two sides. Describe a circle so that the tangents from \(A, B, C\) to it shall each have the same given length.
Through each angular point of a tetrahedron a plane is drawn parallel to the opposite face. Prove that these planes form a second tetrahedron such that each angular point of the first tetrahedron is the centroid of a face of the second tetrahedron.
Prove that the feet of the perpendiculars from any point on the circumcircle of a triangle \(ABC\) on to the sides of a triangle are collinear. Shew that, in general, but one point \(O\) can be found such that the feet of the perpendiculars from \(O\) on four straight lines are collinear. Discuss the exceptional cases.
Shew that the difference of the squares of two tangents to two coplanar circles from any point \(P\) in their plane varies as the perpendicular from \(P\) on their radical axis. On the line joining the centres of similitude of two circles as diameter a circle is described. Shew that if \(P\) be any point on this circle the ratio of the tangents from \(P\) to the other two circles is a constant, one being drawn to each.
Any chord of a circle passes through a fixed point \(O\). Prove that the tangents at the ends of the chord always intersect on a fixed straight line. A circle passing through \(A\) and \(B\) intersects \(CA, CB\) in \(E\) and \(F\). If the middle point of \(EF\) be \(V\), prove that the pole of \(CV\) with respect to the circle \(ABC\) lies on \(AB\).
Prove that the inverse of a circle is a circle or a straight line. Any two circles are drawn cutting each other orthogonally and touching each of two fixed circles. Prove that the points of intersection of the two circles lie on one of four fixed circles.
Shew that tangents to a conic at the extremities of a focal chord intersect on the directrix. The distances of a point \(Q\) from the axis and directrix of a parabola are one half and twice the latus rectum respectively. Prove that, of the normals drawn from \(Q\) to the parabola, two are at right angles.
Shew that the ratio of the rectangles contained by the segments of two intersecting chords of a conic which are parallel to two fixed straight lines is independent of the position of the point of intersection of the chords. \(AP\) is a chord of an ellipse through its vertex \(A\) and a straight line is drawn through \(P\) at right angles to \(AP\) and meets the major axis in \(R\). Prove that the focal chord parallel to \(AP\) is equal to \(AR\).