\(ABC\) is a triangle obtuse-angled at \(A\); \(D\) is the foot of the perpendicular from \(A\) on the side \(BC\). If \(BD\) equals \(3AD\), and \(CD\) equals \(2AD\), prove by calculations based on geometrical theorems that the angle \(BAC\) is \(135^\circ\).
\(A_1, A_2, A_3, A_4\) are four coplanar points such that the line joining any two is perpendicular to the line joining the other two; \(S_1, S_2, S_3, S_4\) are the circumcentres of the triangles \(A_2A_3A_4, A_3A_4A_1, A_4A_1A_2, A_1A_2A_3\). Prove that \(A_1\) is the circumcentre of \(S_2S_3S_4\), and that all the eight triangles whose vertices are either three of the \(A\)'s or three of the \(S\)'s have the same nine point circle. Shew also that the quadrangles \(A_1A_2A_3A_4\) and \(S_1S_2S_3S_4\) are congruent.
\(ABC\) is a triangle, \(O\) the centre of its circumcircle. Forces \(P,Q,R\) act along \(BC, CA, AB\), and forces \(P',Q',R'\) along \(OA, OB, OC\). Shew that if the forces are in equilibrium \[ P\cos A + Q\cos B + R\cos C = 0, \] and \[ \frac{PP'}{\sin A} + \frac{QQ'}{\sin B} + \frac{RR'}{\sin C} = 0. \]
Explain and illustrate the principle of duality in projective geometry, and discuss the bearing on this principle of the theory of reciprocation with respect to a conic.
\(AB\) is a fixed chord of a circle, and \(KL\) is a variable chord of fixed length; \(AK\) and \(BL\) intersect at \(P\). Prove that \(P\) lies on one or other of two fixed circles.
\(p,q,r,s\) are the four common tangents to two conics \(S\) and \(S'\). The points of contact of \(p\) are \(P, P'\); those of \(q, Q, Q'\); \(r\) meets \(p\) and \(q\) in \(A\) and \(B\); \(s\) meets \(p\) and \(q\) in \(C\) and \(D\). Prove that the cross ratios \((PAP'C)\) and \((QDQ'B)\) are equal; and that if \(PP'\) divides \(AC\) harmonically and \(r,s\) meet in \(E\), \(E\) is conjugate to \(P'\) with respect to \(S\) and to \(P\) with respect to \(S'\).
\(ABCDEFGH\) is an octagon composed of eight similar uniform rods, each of weight \(w\), freely hinged together. Shew that it can be supported as a regular octagon in a vertical plane with \(AE\) vertical if certain vertical forces are applied at the angular points other than \(A\) and \(E\); sketch the force diagram and determine these forces.
Investigate the correspondence between points in a plane defined by \[ x' : y' : z' :: a_1x+b_1y+c_1z : a_2x+b_2y+c_2z : a_3x+b_3y+c_3z, \] where the determinant of the coefficients does not vanish. Prove in particular that straight lines correspond to straight lines; that the cross ratio of any four points is equal to that of the four corresponding points; that there are in general only three but there may be a whole line of double points; and that it is possible to assign arbitrarily four pairs of corresponding points, subject to a restriction about collinearity. Prove also that it is possible to choose the coefficients so that any two arbitrarily assigned non-degenerate conics correspond, and reconcile this with the impossibility of choosing arbitrarily five pairs of corresponding points.
Prove that \[ e + \frac{2}{e} = \sum_{n=0}^{\infty} \frac{5n+1}{2n+1}. \]
Shew that if a triangle be self-polar with regard to a rectangular hyperbola its in- and ex-centres lie on the hyperbola.