Prove the harmonic properties of a complete quadrilateral. \(ABCD\) is a quadrilateral, \(AB\) and \(CD\) meet in \(Q\), \(BC\) and \(AD\) meet in \(R\), \(AC\) and \(BD\) in \(P\). Shew that if \(PQ\) meets \(AD\) in \(S\) and \(PR\) meets \(AB\) in \(T\), then \(BS, DT\) and \(AC\) are concurrent and \(ST, BD\) and \(QR\) are concurrent.
Prove that in an ellipse \(SP.S'P = CD^2\), where \(CD\) is the semi-diameter conjugate to \(CP\). Tangents drawn to an ellipse at the extremities of two conjugate semi-diameters intersect in \(P\). Prove that the rectangle contained by the focal distances of \(P\) cannot exceed the sum of the squares of the semi-axes of the ellipse.
Prove that the equations of two circles cutting at right angles may be put in the form \[ x^2+y^2-2cx\cot\theta = c^2 \quad \text{and} \quad x^2+y^2+2cx\tan\theta = c^2. \] Prove that the locus of points at which these circles subtend equal angles is \[ x^2+y^2+2cx\tan 2\theta = c^2. \]
Find the invariants of \(ax^2+2hxy+by^2+2gx+2fy+c\) for a transformation from one set of rectangular axes to another. Prove that the transformation of rectangular axes which converts \(X^2/\alpha+Y^2/\beta\) into \(ax^2+2hxy+by^2\) will convert \(X^2/(\alpha-\theta)+Y^2/(\beta-\theta)\) into \[ \{ax^2+2hxy+by^2-\theta(ab-h^2)(x^2+y^2)\}/\{1-(a+b)\theta+(ab-h^2)\theta^2\}. \]
A circle touches a hyperbola at two points, the chord of contact being parallel to the transverse axis. Prove that the length of the tangent to the circle from any point of the hyperbola is to the distance of the point from the chord of contact as \(e:1\), where \(e\) is the eccentricity of the conjugate hyperbola.
Interpret the equation \(S-\alpha T=0\), where \(S=0\) is a conic, \(T=0\) is the tangent to the conic at a point \(P\) on it and \(\alpha=0\) the equation of a straight line through \(P\). Find the equation of the circle of curvature at the point \(P(ct, c/t)\) on the rectangular hyperbola \(xy=c^2\). If \(\rho\) is the radius of curvature, \(n\) the length of the normal chord of the hyperbola at \(P\), and \(l\) the length of the tangent from the centre of the hyperbola to the circle of curvature at \(P\), prove that \[ n=2\rho=l^2/(3\sqrt{3}c^2). \]
Find the condition that the line \(lx+my+nz=0\) should touch the conic \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0. \] Prove that four conics can be drawn through the vertices of the triangle of reference to touch the two lines \(lx+my+nz=0, l'x+m'y+n'z=0\), and shew that the equations of the chords of contact of the conics with these lines in the four possible cases are \[ x(ll')^{1/2} \pm y(mm')^{1/2} \pm z(nn')^{1/2} = 0. \]
\(A\) is a fixed point outside a given fixed circle, and \(P\) is any point on the circumference. The line \(AF\) perpendicular to \(AP\) meets the tangent at \(P\) in \(F\). If the rectangle \(FAPQ\) is completed, prove that the locus of \(Q\) is a straight line.
Any irregular polygon is circumscribed about a circle. Prove that the perimeter of the polygon bears to the perimeter of the circle the same ratio as the area of the polygon to the area of the circle. Prove also that the same theorem is true for a polyhedron circumscribed about a sphere, if perimeter is replaced by surface, and area by volume.
Prove that the inverse of a system of non-intersecting coaxal circles with respect to a limiting point is a system of concentric circles with the inverse of the other limiting point as centre. \(U\) and \(V\) are two non-intersecting circles and \(A\) is a limiting point of the coaxal system of which they are members. A circle drawn through \(A\) and touching \(V\) at \(F\) meets \(U\) in \(P\) and \(Q\). Prove that \[ \frac{PF}{QF} = \frac{AP}{AQ}. \]