If \(P, Q,\) and \(R\) are points on the sides \(BC, CA,\) and \(AB\) respectively of a triangle \(ABC\) such that \[ \mu.BP+\nu.CP=\nu.CQ+\lambda.AQ=\lambda.AR+\mu.BR=0, \] shew that the lines \(AP, BQ\) and \(CR\) meet in a point \(O\) and that \[ \lambda.Al + \mu.Bl + \nu.Cl = (\lambda+\mu+\nu).Ol, \] where \(l\) is any line of the plane and \(Al\), for instance, denotes the perpendicular distance of \(A\) from \(l\), regard being had to sign. Hence shew that, if \(I, I_1, I_2\) and \(I_3\) are the centres of the inscribed and escribed circles of the triangle, \[ (s-a).I_1l + (s-b).I_2l + (s-c).I_3l = s.Il. \]
Shew that if \(A, B, C\) and \(A', B', C'\) are sets of points on two coplanar lines, then the points of intersection \((BC', B'C), (CA', C'A),\) and \((AB', A'B)\) are collinear. Three lines \(a, b,\) and \(c\) meet in a point \(I\), and \(d\) is any line not passing through \(I\). Two ranges of points \((A_1, A_2, \dots)\) and \((B_1, B_2, \dots)\) on \(a\) and \(b\) respectively are in perspective from a point \(O\). Shew that two points \(H\) and \(K\) can be found on \(c\) such that the ranges \((A_1, A_2, \dots)\) and \((B_1, B_2, \dots)\) are in perspective with the same range on \(d\), the centres of perspective being \(H\) and \(K\) respectively.
Define an involution of points on a line and shew that an involution is determined by two pairs of points. Chords joining two fixed points \(A\) and \(B\) of a conic to a variable point \(P\) of the conic meet any line \(l\) in pairs of points. Shew that these form an involution only if \(l\) goes through the pole of \(AB\). Shew that if four coplanar circles meet in a point \(O\) the six lines which join \(O\) to the other intersections of the circles are three pairs of lines of an involution pencil.
Shew that the polar reciprocal of a conic with respect to a circle whose centre is at a focus is a circle. Shew that the reciprocals of a system of confocal conics with respect to a circle whose centre is one of the common foci are circles of a coaxal system. How do the limiting points arise, and what happens to the property that two confocal conics cut at right angles?
Shew that there are two spheres which touch a given right circular cone along circles and also touch a given plane \(\alpha\). The points of contact of the spheres with \(\alpha\) are \(S\) and \(S'\). The line \(SS'\) meets the cone in \(A\) and \(A'\) and the planes of the circles of contact in \(K\) and \(K'\). Shew that \(A\) and \(A'\) are harmonically separated by \(S\) and \(K\) and also by \(S'\) and \(K'\) and that if \(P\) is any point on the section of the cone by the plane \(\alpha\), then either \[ SP+S'P = AA', \] or \[ SP \sim S'P = AA'. \]
Find equations for the incentre of the triangle formed by the lines \[ x-2y=0, \quad 4x-3y+5=0, \quad x+3y-10=0, \] and obtain the coordinates of the orthocentre.
A line \(l\) meets the parabola \(y^2=4ax\) in \(P\) and \(Q\). The line through \(P\) parallel to the tangent at \(Q\) meets the line through \(Q\) parallel to the tangent at \(P\) in a point \(R\). If \(l\) varies, passing through a fixed point \((\alpha, \beta)\), shew that the locus of \(R\) is the parabola \[ 2y^2 - 2ax - 3\beta y + 6a\alpha = 0. \]
Shew that the equations of the chords of contact of any conic \(S\) which has double contact with each of two given conics \(S_1\) and \(S_2\) are of the form \(\lambda\alpha+\mu\beta=0\) and \(\lambda\alpha-\mu\beta=0\), where \(\alpha=0\) and \(\beta=0\) are lines joining the points of intersection of \(S_1\) and \(S_2\) in pairs. Conversely for any value of \(\frac{\lambda}{\mu}\) a conic can be found having double contact with \(S_1\) along \(\lambda\alpha+\mu\beta=0\) and with \(S_2\) along \(\lambda\alpha-\mu\beta=0\). Shew that there are six conics \(S\) which pass through a given point, and six conics \(S\) such that the chords of contact are conjugate with respect to \(S\).
Shew that the locus of poles of a line \(l\) with respect to the conics of a confocal system is a line perpendicular to \(l\), and that this is the normal to the conic of the system which touches \(l\) at its point of contact with \(l\). Hence shew that if tangents are drawn from \((x_1, y_1)\) to the conic \[ \frac{x^2}{a^2+\lambda} + \frac{y^2}{b^2+\lambda} = 1, \] the normals at the points of contact will intersect at the point \((\xi, \eta)\) given by \[ \frac{\xi}{x_1(b^2+\lambda-y_1^2)} = \frac{-\eta}{y_1(a^2+\lambda-x_1^2)} = \frac{a^2-b^2}{(a^2+\lambda)y_1^2 + (b^2+\lambda)x_1^2}. \] Deduce that the locus of \((\xi, \eta)\) as \(\lambda\) varies is a straight line.
Find the equation of the line \(l\) joining the points of intersection of \(x = \lambda y\) and \(x = \mu z\) with the conic \[ ayz+bzx+cxy=0 \] other than the vertices of the triangle of reference. If \(\lambda\) and \(\mu\) are allowed to vary subject to the condition \[ a\lambda\mu + \beta\lambda + \gamma\mu + \delta = 0, \] shew that the line \(l\) will pass through a fixed point provided that \(a\alpha = c\beta+b\gamma\), and find the coordinates of the point.