Prove that the polars of the points of a circle \(C\) with respect to a non-concentric circle \(D\) envelop a conic \(\Sigma\), one of whose foci is the centre of \(D\). Chords of a conic \(S\) are drawn subtending a right angle at a fixed point \(K\). Prove that they envelop a conic of which \(K\) is a focus.
Of the four coplanar points \(A, B, C, E\) no three are collinear; \(AE\) intersects \(BC\) in \(L\); \(BE\) intersects \(CA\) in \(M\); \(CE\) intersects \(AB\) in \(N\). A conic \(S\) through \(L, M, N\) cuts \(BC, CA, AB\) again in \(P, Q, R\) respectively. Prove that \(AP, BQ, CR\) are concurrent (at, say, \(U\)). If \(S\) varies subject to the additional condition of passing through a fourth fixed point \(K\), prove that the locus of \(U\) is a conic through \(A, B, C\).
We define \[ S = ax^2+by^2+cz^2+2fyz+2gzx+2hxy, \] \[ l_i = p_ix+q_iy+r_iz, \quad i=1, 2. \] Interpreting \(x,y,z\) as homogeneous co-ordinates of a point in a plane, prove that, for any constant \(\lambda\), \[ S - \lambda l_1l_2 = 0 \] is a conic through the points in which \(l_1=0\) and \(l_2=0\) cut \(S=0\). If there are exactly four of these points, prove that any conic (with one exception) through these four points can be so expressed for a suitable value of \(\lambda\). Interpret the equation \(S- \lambda l_1^2=0\), separating the case in which \(l_1=0\) is a tangent to \(S=0\) from the general case. Prove that, if \(P\) is a point on \(l_1=0\), the points of contact of the tangents from \(P\) to \(S-\lambda l_1^2=0\) and to \(S=0\) are collinear.
\(ABCD\) is a square of side \(a\). A point \(P\) moves so that the sum of the squares of its distances from \(A, B, C, D\) is equal to \(3a^2\). Prove that its locus is a circle and find the radius.
Given one vertex \(A\), the circumcentre \(O\), and the orthocentre \(H\) of a triangle, show how to construct the triangle. Two triangles \(ABC\) and \(AB'C'\) with common vertex \(A\) are such that the orthocentre of each is the circumcentre of the other. Prove that the point of intersection of \(BC\) and \(B'C'\) is equidistant from the circumcentre and orthocentre.
If a circle \(S\) when inverted with respect to a circle \(\Sigma\) becomes a circle \(S'\), show that \(S, \Sigma,\) and \(S'\) are members of a coaxial system. Prove that for any triangle the circumcircle, the nine-points circle, and the polar circle (that is, the circle with respect to which the triangle is self-polar) are coaxial.
Establish Pascal's theorem that if a hexagon is inscribed in a conic then the meets of the three pairs of opposite sides are collinear. State the dual theorem. Prove that if a tangent to a hyperbola at a point \(P\) meets the asymptotes in \(A\) and \(B\), then \(AP=PB\).
\(P\) and \(Q\) are two points on an ellipse of which \(S\) is a focus and are such that \(\angle PSQ\) is always a constant angle. Show that the tangents at \(P\) and \(Q\) meet on a fixed conic, and that the chord \(PQ\) envelops a second fixed ellipse. In what circumstances can the former conic be a hyperbola?
Prove that the two pairs of lines \(ax^2+2hxy+by^2=0\) and \(a'x^2+2h'xy+b'y^2=0\) are harmonically conjugate if \(ab'+a'b=2hh'\). In a given plane \(A, B\) are two fixed points and \(l, m\) two fixed lines. A point \(P\) is such that \(PA, PB\) harmonically separate the lines through \(P\) parallel to \(l, m\). Show that the locus of \(P\) is a hyperbola whose asymptotes are parallel to \(l, m\), and that its centre lies on \(AB\).
\(P\) is a point on a conic \(S=0\) and the tangent at \(P\) has equation \(T=0\) while \(L=0\) represents any line through \(P\). Interpret the equations: