Shew that an ellipse can be orthogonally projected into a circle. The four common tangents to two similar and similarly situated ellipses and the line of collinearity of the middle points of these tangents all touch the same parabola.
Prove that the polar reciprocal of a conic with respect to a point on the conic is a parabola. Apply this proposition to shew that, if a variable chord of a conic subtend a right angle at a fixed point on the conic, then the chord always passes through a fixed point.
Shew that a pencil of four rays cuts any transversal in a range of constant anharmonic ratio. Extend this theorem to the case of the points of section of a transversal by four planes having a common line of intersection.
Prove that the locus of a point \(P\), which moves in a plane so that the ratio of its distances from two fixed points \(A\) and \(A'\) in the plane are constant, is a circle. Shew also that, if the tangent at \(P\) to the circle cuts \(AA'\) in \(T\), and the straight line \(BTB'\) drawn through \(T\) at right angles to \(PT\) cuts \(PA\) and \(PA'\) in \(B\) and \(B'\) respectively, the four points \(A, A', B, B'\) are concyclic.
Define the radical axis of two circles and shew how to construct it for two circles which do not intersect. Find a construction for a circle which shall pass through two fixed points and cut a given circle orthogonally.
If three concurrent straight lines drawn from the angular points \(A, B, C\) of a triangle cut the opposite sides in \(D, E, F\) respectively, prove that \[ AE \cdot CD \cdot BF = AF \cdot BD \cdot CE. \] Prove also that, if \(EF\) be produced to cut \(BC\) in \(G\), \((BC, DG)\) is an harmonic range.
Prove that, if a straight line be at right angles to two intersecting straight lines, it will be at right angles to every other straight line in their plane. If the three angles forming one corner of a tetrahedron are right angles, prove that the perpendicular from that corner on the opposite triangular face intersects it at its orthocentre.
The tangents drawn from a point \(P\) to a parabola whose focus is \(S\) touch it at \(Q\) and \(Q'\); prove that \(SP^2=SQ \cdot SQ'\). Prove that the locus of the focus of a parabola which touches the three sides of a given triangle is a circle.
Prove that in an ellipse the locus of the middle points of parallel chords is a straight line. \(CP, CD\) are conjugate semi-diameters of an ellipse, the tangent at \(P\) meets the major axis in \(T\), and \(N\) is the foot of the ordinate of \(P\). If \(PD\) meets the major axis in \(K\), prove that \[ KD : KP :: CN : NT. \]
The tangent at any point \(P\) of an hyperbola, whose foci are \(S\) and \(S'\), cuts one asymptote in \(L\), and \(SP\) produced cuts the same asymptote in \(R\); prove that \(SR=RL\). Prove also that if the tangent cut the other asymptote in \(L'\), the four points \(S, S', L, L'\) are concyclic.