Given a triangle \(ABC\), points \(Q\), \(M\) are taken on the side \(AC\) such that \(AQ = \frac{1}{4}AC\), \(AM = \frac{2}{3}AC\), and points \(R\), \(N\) are taken on the side \(AB\) such that \(AR = \frac{1}{4}AB\), \(AN = \frac{3}{4}AB\). The line \(BQ\) meets \(CR\), \(CN\) in \(X\), \(Y\) respectively; the line \(BM\) meets \(CR\), \(CN\) in \(U\), \(T\) respectively. Prove that \(XY\) bisects \(BC\) and that \(UV\) is parallel to \(BC\).
Three points \(A\), \(B\), \(C\) are given in general position in a plane. A circle of the coaxal system with \(A\), \(B\) as limiting points meets a circle of the coaxal system with \(A\), \(C\) as limiting points in \(U\), \(V\). Prove that the line \(UV\) passes through the circumcentre of the triangle \(ABC\).
Three concurrent lines \(DA\), \(DB\), \(DC\) in space are such that each is perpendicular to the other two. Identify the common chord of the three spheres with diameters \(BC'\), \(CA\), \(AB\).
The ellipse \(x^2/a^2 + y^2/b^2 = 1\) has foci \(S(ae, 0)\) and \(S'(-ae, 0)\); \(P(x_1, y_1)\), where \(x < ae\), is an arbitrary point of the ellipse (taken, for convenience, in the first quadrant) and \(Q\) is the point \((x_1, -y_1)\). Prove that a circle can be drawn to touch each of the segments \(SP\), \(S'P\), \(SQ\), \(S'Q\). Find the centre of the circle, and prove that its radius is \(ey_1\).
Prove that the chords of the parabola \(y^2 = 4ax\) which subtend a right-angle at the origin \(O\) all pass through the point \((4a, 0)\). The line through the point \((16a, 0)\) perpendicular to the \(x\)-axis meets the parabola in \(U\), \(V\). Prove that, if \(P\) is any point of the parabola on the side of \(UV\) remote from \(O\), then the circle on \(OP\) as diameter meets the parabola again in two (real) points, and that the line joining those two points passes through the fixed point \((-4a, 0)\).
A chord \(PQ\) of a rectangular hyperbola meets the asymptotes at \(U\), \(V\). Prove that \(PU = QV\). If the point \(U\) is given on one of the asymptotes, give a geometrical construction for the chord \(PQ\) passing through \(U\) such that \(PQ = \frac{1}{2}UV\). For a particular position of \(U\), the circle on the resulting chord \(PQ\) as diameter touches the asymptote through \(U\). Prove that \(PQ\) then makes an angle of \(\frac{1}{4}\pi\) with that asymptote.
An ellipse has equation \(x^2/a^2 + y^2/b^2 = 1\). The parabola is drawn with focus \(A'(-a, 0)\) and directrix the line through \(A'(-a, 0)\) perpendicular to the axis, and \(P\) is one of the points on the parabola such that the tangents from \(P\) to the ellipse are perpendicular. The tangent at \(P\) to the parabola meets \(AA'\) in \(T\) and the foot of the perpendicular from \(P\) is on \(AA'\). Prove that, if \(T\) is a point of trisection of \(AA'\), then so is \(N\), and find the value of \(b/a\) in this case.
Define the cross-ratio of four points \(P\), \(Q\), \(R\), \(S\) on a line \(l\), and prove from your definition that, if \(P'\), \(Q'\), \(R'\), \(S'\) are four points on another line \(l'\) such that \(PP'\), \(QQ'\), \(RR'\), \(SS'\) meet through \(Y\) and such that the cross-ratios \((PQRS)\), \((P'Q'R'S')\) are equal, then \(SS'\) also passes through \(Y\). \(A\), \(B\), \(C\), \(D\) are four points in general position in a plane; \((BC\), \(AD)\), \((CA\), \(BD)\), \((AB\), \(CD)\) meet in points \(X\), \(Y\), \(Z\) respectively. An arbitrary line through \(Y\) meets \(AD\) in \(U\) and \(BC\) in \(V\); \(ZU\) meets \(AD\) in \(M\) and \(ZV\) meets \(BC\) in \(N\). Prove that the line \(MN\) passes through \(X\). Deduce a theorem for a parallelogram \(ABCD\), and prove it independently.
\(ABC\) is a given triangle and \(l\) a given line in its plane. A variable conic is drawn touching \(AB\), \(AC\) at \(B\), \(C\) respectively and meeting \(l\) at \(M\), \(N\). Prove that the tangents at \(M\), \(N\) meet on a certain fixed straight line. Two other lines are obtained similarly after cyclic permutation of the letters \(A\), \(B\), \(C\). Prove that the three lines so obtained are concurrent.
The polar of the point \(D(1, 1, 1)\) with respect to the conic whose equation (in homogeneous coordinates with triangle of reference \(XYZ\)) is $$fyz + gzx + hxy = 0$$ meets the conic in \(P_1\), \(P_2\), and the lines \(XP_1\), \(XP_2\) meet \(YZ\) in \(L_1\), \(L_2\). Points \(M_1\), \(M_2\) on \(ZX\) and \(N_1\), \(N_2\) on \(XY\) are defined similarly. Prove that the six points \(L_1\), \(L_2\), \(M_1\), \(M_2\), \(N_1\), \(N_2\) all lie on the conic $$gh(y + h)x^2 + hf(h + f)y^2 + fg(f + g)z^2 + 2fgh(yz + zx + xy) = 0.$$