The vertices \(P\), \(Q\), \(R\) of a triangle \(PQR\) lie on the sides \(BC\), \(CA\), \(AB\) respectively of a fixed triangle \(ABC\). Prove that the circles \(AQR\), \(BRP\), \(CPQ\) meet in a point \(K\), and that all such inscribed triangles \(PQR\) which give rise to the same point \(K\) are similar. Outline a construction for an inscribed equilateral triangle \(PQR\) whose vertex \(P\) is given on \(BC\).
\(A\), \(B\), \(C\), \(D\) are four points on a circle \(S\). \(BC\) and \(AD\) meet in \(X\), \(CA\) and \(BD\) meet in \(Y\). Prove that the centre of \(S\) is the orthocentre of the triangle \(XYZ\), and that the circles \(ABX\), \(ADZ\) meet again in the foot of the perpendicular from \(Y\) on \(ZX\).
Two points \(P\), \(Q\) invert into the points \(P'\), \(Q'\) with respect to a circle with centre \(O\) and radius \(k\). Prove that $$\frac{P'Q'}{PQ} = \frac{k^2}{OP \cdot OQ}.$$ \(A\), \(B\), \(C\), \(D\) are any four points in a plane. Prove that $$AB \cdot CD + AD \cdot BC \geq AC \cdot BD,$$ equality occurring only when \(A\), \(B\), \(C\), \(D\) lie, in the order \(ABCD\), on a circle or straight line.
Prove that the common chords of a central conic and a circle taken in pairs are equally inclined to the principal axes of the conic. The common chord of the ellipse $$\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$$ and the circle of curvature at a point \(P\) passes through the point \((\xi, \eta)\). Prove that there are four such points \(P\) and that they lie on the circle $$2(x^2 + y^2) - (a^2 - b^2)(\xi x/a^2 - \eta y/b^2) - a^2 - b^2 = 0.$$
Prove that three normals can be drawn to a parabola \(\Gamma\) from a general point, and that the circle through the feet of the normals passes through the vertex of \(\Gamma\). \(P_0\), \(P_1\), \(P_2\), \(\ldots\) is a sequence of points such that, for each \(i \geq 1\), \(P_i\) is the centre of the circle through the feet of the three normals to \(\Gamma\) from \(P_{i-1}\). Prove that, as \(n\) tends to infinity, the point \(P_n\) tends to a limiting position \(H\) which is independent of \(P_0\). Identify the point \(H\).
At each point of a parabola is drawn the rectangular hyperbola of four-point contact. Prove that the locus of its centre is the reflexion of the parabola in its directrix.
Sketch the locus (the cycloid) given by $$x = a(t + \sin t), \quad y = a(1 + \cos t),$$ for values of \(t\) between \(0\) and \(4\pi\), where \(a > 0\). Find the co-ordinates of the centre of curvature at the point \(t\), and prove that the locus of the centre of curvature is an equal cycloid; illustrate this in your diagram.
A convex polyhedron \(P\) has, for its faces, \(x\) triangles and \(y\) (convex) quadrilaterals, where \(x\) and \(y\) are both positive. The faces meeting in a vertex of \(P\) consist of \(a\) triangles and \(b\) quadrilaterals, where \(a\) and \(b\) are the same for all vertices. Prove that $$\frac{3x}{a} = \frac{4y}{b} = \frac{1}{2}(x + 2y + 4),$$ and find all the positive integral values of \(a\), \(b\), \(x\), \(y\) that satisfy these equations. Give examples of polyhedra satisfying these conditions when \(b = 2\).
\(ABC\), \(A'B'C'\) are two skew lines, and \(AB:BC = A'B':B'C'\). Prove that the mid-points of \(AA'\), \(BB'\), \(CC'\) are collinear.
Show that the cubic curve whose equation in rectangular Cartesian co-ordinates is $$x^3 - x^2y - 2xy^2 + 5xy + 2y^2 = 0$$ has a double point at the origin. Find the equations of the asymptotes and the co-ordinates of their finite points of intersection with the curve. Give a sketch of the curve.