Find the equation of the circumcircle of the triangle whose sides are the line \(lx+my+n=0\) and the pair of lines \(ax^2+2hxy+by^2=0\). Interpret the result geometrically when \(am^2-2hlm+bl^2=0\).
\(P\) is any point of the parabola \[ y^2=a(x-a), \] and \(O\) is the vertex of the parabola \[ y^2=4ax. \] The circle on \(OP\) as diameter meets the second parabola in three points other than \(O\). Prove that the normals at these three points meet in a point of the parabola \[ y^2=4a(x+a). \]
\(A, B, C\) are three points of a conic and the tangents at \(B\) and \(C\) meet in \(A'\). The points \(B'\) and \(C'\) are similarly defined. Prove that the lines \(AA', BB', CC'\) are concurrent. Deduce theorems for a hyperbola and a parabola by taking the line at infinity (i) as the line \(BC\), (ii) as the line \(B'C'\).
Interpret the equations \[ (i) \ S-\lambda u^2=0, \quad (ii) \ S-\mu uv=0, \] where \(S=0\) is the equation of a conic, \(u=0\) is a general line, \(v=0\) is the tangent to \(S\) at one of the intersections of \(S=0\) and \(u=0\), and \(\lambda\) and \(\mu\) are constants. A conic \(S\) touches the sides \(AB, AC\) of a triangle \(ABC\) at \(B\) and \(C\) respectively. Conics \(S_1, S_2\) pass through \(B\) and \(C\) and have three-point contact with \(S\) at \(B\) and \(C\) respectively. The line \(l\) which joins the other two intersections of \(S_1\) and \(S_2\) meets \(S\) in the points \(H, K\). Prove that there exists a conic which touches \(S\) at \(H\) and \(K\) and for which the triangle \(ABC\) is self-polar.
Two pairs of opposite edges of a tetrahedron are perpendicular. Prove that the third pair are perpendicular and that the perpendiculars from the vertices to the opposite faces are concurrent. If further the three lines joining the mid-points of pairs of opposite edges are mutually perpendicular, show that the tetrahedron is regular.
Define a homography on a straight line \(l\). Under a given homography \(T\) on \(l\) the points \(P, Q\) correspond to \(P', Q'\) respectively and the point \(A\) is self-corresponding. \(P'', Q''\) are arbitrary points on another line \(m\) through \(A\); the lines \(PP'', QQ''\) meet in \(U\) and the lines \(P'P'', Q'Q''\) meet in \(V\). Prove that the line \(UV\) meets \(l\) in a self-corresponding point of \(T\). If only the points \(A, P, P', Q\) are given, but it is known in addition that \(T\) has one and only one self-corresponding point (at \(A\)), construct the point \(Q'\).
\(A, B, C, D\) are four coplanar points in general position. A line \(l\) meets \(BC, CA, AB, DA, DB, DC\) in \(P, Q, R, P', Q', R'\) respectively. By considering a range of points on the line \(BC\), or otherwise, prove that the cross-ratios \((PP'Q'R')\) and \((P'PQR)\) are equal, and deduce that \((P, P'), (Q, Q'), (R, R')\) are pairs of an involution on \(l\).
Sketch the curve whose equation in Cartesian coordinates is \[ y^4+axy^2+a^2x^2=a^4, \] where \(a\) is a positive constant. Show that the curve may be inscribed in a certain rectangle, of area \(2\{(\frac{5}{4})^{\frac{1}{2}}+(\frac{5}{4})^{\frac{3}{4}}\}a^2\), which touches the curve at five points.
The vertex \(A\) of a triangle is at a fixed point of a given circle with centre \(O\). The base \(BC\) is a variable chord of the circle and passes always through a fixed point \(P\). Show that the locus of the orthocentre of \(ABC\) is a circle, and indicate its centre and radius in a sketch.
Assign a geometrical meaning to the expression \(x^2+y^2+2gx+2fy+c\). Establish the existence of a radical axis for a pair of circles. Show that the radical axes of three circles taken in pairs are concurrent.