Prove Menelaus' theorem, that if a transversal meets the sides \(BC, CA, AB\) of a triangle \(ABC\) in \(D, E, F\) respectively, then \(AF.BD.CE = -FB.DC.EA\). Show that the tangents to the circumcircle of a triangle at its vertices meet the opposite sides in three collinear points.
Explain what is meant by inversion in geometry, and show that the inverse of a circle is either a circle or a straight line. The inverses with regard to a given circle \(\gamma\) of two points \(P, Q\) are two points \(P', Q'\) respectively, and \(C\) is any point on the circle of inversion. Show that the circumcircles of the triangles \(PCQ\) and \(P'CQ'\) meet on \(\gamma\).
Prove that the tangents drawn to a circle from a given external point are equal. The sides of a skew (non-planar) quadrilateral \(ABCD\) each touch a sphere at the four points \(P, Q, R,\) and \(S\). Prove that the quadrilateral \(PQRS\) is cyclic.
Show that if \(S=0\) and \(S'=0\) represent the cartesian equations of two circles, then \(S+kS'=0\) also represents a circle, and explain its relationship to the first two circles. If the tangents from two given points to a variable circle are of given lengths, prove that the variable circle always passes through two fixed points, and state the positions of these two points.
Prove that the locus, as \(t\) varies, of the point whose rectangular coordinates are given by \[ x=at^2+2bt+c, \quad y=a't^2+2b't+c' \] is a parabola. Find the equation of the tangent at the point \(t\), and show that the tangents at the points \(t_1, t_2\) meet at the point \[ at_1t_2+b(t_1+t_2)+c, \quad a't_1t_2+b'(t_1+t_2)+c'. \] Show that the directrix of the parabola is \[ ax+a'y+b^2+b'^2-ac-a'c'=0. \]
A conic is inscribed in a triangle \(ABC\), and \(D\) is its point of contact with \(BC\). The tangent parallel to \(BC\) meets \(AB\) and \(AC\) in \(L\) and \(M\) respectively and touches the conic at \(N\). By considering the cross-ratios of the ranges produced by the four tangents \(BC, CA, AB\) and \(LM\) on \(BC\) and \(LM\), prove that \(BC:DC = LM:LN\). Hence, or otherwise, prove that if an ellipse inscribed in a triangle has its centre at the circumcentre then the three altitudes are normals to it.
Show that the polar equation of a conic referred to a focus as pole may be put in the form \[ l/r = 1+e\cos\theta, \] and find the equation of the tangent at the point \(\theta=\alpha\). A conic has foci \(S\) and \(H\), and the tangents from a general point \(R\) touch the conic at \(P\) and \(Q\). Prove that \(RS\) and \(RH\) bisect the angles \(PSQ\) and \(PHQ\) respectively.
Find the condition, or conditions, that the general equation of the second degree \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \] should represent (i) a parabola, (ii) a pair of straight lines, and (iii) a pair of coincident straight lines. If the coordinate axes are rectangular, find also the condition for the equation to represent a rectangular hyperbola, and use the condition to prove the theorem that if a rectangular hyperbola circumscribes a triangle it passes through the orthocentre.
Show by comparison with the identity \(4\cos^3\alpha - 3\cos\alpha - \cos 3\alpha = 0\) that the cubic equation \(x^3-3qx-r=0\) can be solved in terms of cosines provided that \(4q^3 > r^2\). If \(\alpha\) is defined by the equation \(\cos 3\alpha = r/2q^{3/2}\), show that \(2q^{1/2}\cos\alpha\) is a root, and find the other two roots. Use the method to solve the equation \[ x^3 - 6x^2 + 6x + 8 = 0. \]
If two triangles \(ABC\) and \(A_1B_1C_1\) are of equal area, prove that \[ \sum_{a,b,c} a^2 \cot A_1 = \sum_{a_1,b_1,c_1} a_1^2 \cot A. \]