\(ABC\) is a triangle in which the angles \(ABC, ACB\) are each equal to twice the angle \(BAC\). Prove that \(AB^2 = BC^2 + AB \cdot BC\). Hence show how to inscribe geometrically a regular pentagon in a given circle.
Prove that the orthocentre \(H\), the centroid \(G\) and the centre \(O\) of the circumcircle of a triangle are collinear, and that \(HG=2GO\). The ends of any diameter \(BC\) of a given circle are joined to a fixed point \(A\) in the plane of the circle; find the locus of the orthocentre of the triangle \(ABC\) and deduce the locus of the circumcentre.
Explain what is meant by a centre of similitude. Prove that two circles have two centres of similitude, and that the circle on the line joining them as diameter is coaxial with the given circles. Given three circles \(S_1, S_2, S_3\), prove that the join of a centre of similitude of \(S_1, S_2\) to a centre of similitude of \(S_1, S_3\) passes through a centre of similitude of \(S_2, S_3\).
Show that in general two spheres can be inscribed in a right circular cone to touch a given plane not passing through the vertex, and that the plane cuts the cone in a conic whose foci are the points of contact of the spheres with the plane. When is the conic a parabola? Show that the foci of parabolic sections of a right circular cone lie upon another right circular cone.
Explain what is meant by a projective correspondence (or homography) between the points on a straight line, and show that there is a unique projective correspondence in which to given points \(A, B, C\) there correspond given points \(A', B', C'\). \(M\) is a self-corresponding point of a projective correspondence on a line and to \(A, B\) there correspond respectively \(A', B'\). On any line through \(M\) two points \(S, S'\) are taken; \(SA, S'A'\) meet in \(P\) and \(SB, S'B'\) in \(Q\). Prove that the point \(N\) in which \(PQ\) meets the given line is the other self-corresponding point. Deduce, or prove otherwise, that if in a projective correspondence the two self-corresponding points coincide in \(M\) (a parabolic correspondence), and to \(A\) corresponds \(A'\) and to \(A'\) corresponds \(A''\), then \(M, A'\) are harmonically separated by \(A, A''\). (Assume the harmonic property of the quadrilateral.)
(i) Find the angle between the straight lines given by the equation (in rectangular Cartesian coordinates) \[ ax^2+2hxy+by^2=0. \] (ii) Show that the equation of the base of the triangle of which these lines are the two sides and whose orthocentre is the point \((x_0, y_0)\) is \[ (a+b)(xx_0+yy_0) = bx_0^2-2hx_0y_0+ay_0^2. \]
Prove that the straight line \[ ty = x+at^2 \] touches the parabola \(y^2=4ax\), and find the coordinates of the point of contact. Prove that the locus of the point of intersection of tangents to the parabola which intercept a fixed length \(l\) on the directrix is \[ (x+a)^2(y^2-4ax)=l^2x^2. \]
Find the condition that the straight lines \(l_1x+m_1y=1\), \(l_2x+m_2y=1\) should be conjugate (i.e. each pass through the pole of the other) with regard to the ellipse \[ x^2/a^2+y^2/b^2=1. \] Two points \(P, Q\) of the ellipse \(x^2/\alpha^2+y^2/\beta^2=1\) are such that the tangents at \(P\) and \(Q\) are conjugate with regard to the ellipse \(x^2/a^2+y^2/b^2=1\). Prove that the chord \(PQ\) touches the ellipse \[ \frac{x^2}{\alpha^4(b^2+\beta^2)} + \frac{y^2}{\beta^4(a^2+\alpha^2)} = \frac{1}{a^2\beta^2+b^2\alpha^2}. \]
The lengths of the semi-axes of an ellipse are \(\alpha, \beta\) (\(\alpha > \beta\)), its centre is at the origin of coordinates and its major axis makes an angle \(\phi\) with the \(x\)-axis. Find its equation. Prove that if two concentric ellipses touch one another the angle \(\theta\) between their major axes is given by \[ \tan^2\theta = \frac{(a^2-a^2)(b^2-\beta^2)}{(a^2-\beta^2)(\alpha^2-b^2)}, \] where \(a, b\) and \(\alpha, \beta\) are their semi-axes.
Find the condition that the straight line joining the two points \(P, Q\), whose homogeneous coordinates are \((x_1, y_1, z_1)\) and \((x_2, y_2, z_2)\), should meet the conic \(\alpha x^2+\beta y^2+\gamma z^2=0\) in two points which are harmonically separated by \(P, Q\). A line \(lx+my+nz=0\) is such that it meets the two conics \(ax^2+by^2+cz^2=0\), \(\alpha x^2+\beta y^2+\gamma z^2=0\) in two pairs of points which are harmonically separated. Prove that \[ l^2(b\gamma+c\beta) + m^2(c\alpha+a\gamma) + n^2(a\beta+b\alpha) = 0. \]