Points \(D, E\), and \(F\) are taken on the sides \(BC, CA,\) and \(AB\), respectively, of the triangle \(ABC\). Shew that the circles \(AEF, BFD,\) and \(CDE\) meet in a point \(K\), and that if \(D, E,\) and \(F\) are collinear \(K\) lies on the circle \(ABC\).
Prove that the inverse of a circle with regard to a point in its plane is either a circle or a straight line. \(P\) is a variable point in the plane of the triangle \(ABC\). If \(PA^2.BC^2 = PB^2.CA^2 + PC^2.AB^2\), shew that the locus of \(P\) is the circle through \(B\) and \(C\) orthogonal to the circle \(ABC\).
Two unequal circles, lying in different planes, meet in two points, \(A\) and \(B\). Shew that there is one sphere which contains the two circles. The tangent planes to this sphere at \(A\) and \(B\) meet in a line \(l\). Shew that there are two points on \(l\) from which the circles are in perspective.
Prove the projective property of the cross ratio, namely that if four lines through a point \(O\) are met by two other lines in the sets of points \(A, B, C, D\) and \(A', B', C', D'\), then the cross ratios \((ABCD)\) and \((A'B'C'D')\) are equal. \(A\) and \(B\) are conjugate points with regard to a conic, and the polars of \(A\) and \(B\) meet in \(C\). \(RP\) is a chord of the conic through \(B\). \(RC\) meets the conic again in \(Q\). Prove that \(PQ\) passes through \(A\).
\(S\) is a given conic and \(P\) and \(Q\) are given points. Prove that pairs of conjugate lines through \(P\) and \(Q\) meet the polar of \(P\) in pairs of points in involution. Hence shew that the locus of the point of intersection of pairs of conjugate lines through \(P\) and \(Q\) is a conic \(S'\), which passes through \(P\) and \(Q\), and also that the line \(PQ\) has the same pole with regard to the conics \(S\) and \(S'\). Deduce that the locus of points of intersection of pairs of perpendicular tangents to any conic is a concentric circle.
Find the equation of the circle through the feet of the three normals to the parabola \(y^2 = 4ax\) which pass through the point \((\alpha, \beta)\). Shew that, if \((\alpha, \beta)\) moves along a fixed line, the corresponding circles form a coaxal system.
\(S=0\) is a conic, and \(l=0, l'=0\) are two lines. Interpret the equation \(S+\lambda ll' = 0\), where \(\lambda\) is a number. Shew that the circle of curvature of the rectangular hyperbola \(xy=c^2\) at the point \((ct, \frac{c}{t})\) meets the hyperbola again in the point \((\frac{c}{t^3}, ct^3)\), and find the coordinates of the centre of curvature.
Prove that the poles of the line \(p\), whose equation is \(lx+my+1=0\), with regard to the conics of a confocal system \(\frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda} = 1\) lie on a line \(q\). Prove further that, if \(p\) passes through a fixed point \((\alpha, \beta)\), \(q\) touches the parabola, whose equation in tangential coordinates is \[ (a^2-b^2)lm = \beta l - \alpha m. \] Shew that this parabola is also the envelope of the polars of \((\alpha, \beta)\) with regard to the conics of the confocal system.
The lines \(y=mx, y=m'x\) meet a conic through the origin in the points \(P\) and \(Q\). If \(mm'=K\) (a constant), shew that the line \(PQ\) passes through a fixed point. If \(K=-1\), shew that this fixed point lies on the normal to the conic at the origin.
(i) Use homogeneous coordinates to prove that, if two triangles are in perspective, their corresponding sides meet in collinear points. (ii) Shew that the lines joining the vertices of the triangle of reference to the points of intersection of the opposite sides with the conic \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0 \] touch a conic whose equation in tangential coordinates is \[ bcl^2+cam^2+abn^2=2afmn+2bgnl+2chlm. \]