Prove Pappus's theorem that, if \(A, B, C\) and \(P, Q, R\) are two triads of collinear points on (distinct) coplanar lines, then the three points of intersection \((BR, CQ), (CP, AR), (AQ, BP)\) are collinear. Prove further that, if the triads \(A, B, C\) and \(P, Q, R\) are on two skew lines, and if \(O\) is an arbitrary point of space, then the transversals from \(O\) to the three line-pairs \((BR, CQ), (CP, AR), (AQ, BP)\) are coplanar.
The rectangular hyperbola \(xy=c^2\) meets the ellipse \(b^2x^2+a^2y^2=a^2b^2\) in four real points \(A, B, C, D\). Prove that the area of the parallelogram \(ABCD\) is \(2\sqrt{(a^2b^2-4c^4)}\).
Prove that the equations of two given conics through four distinct points can be expressed in terms of general homogeneous coordinates in the form \begin{align*} ax^2+by^2+cz^2 &= 0, \\ x^2+y^2+z^2 &= 0. \end{align*} Prove that the pole, with respect to any conic through the four points, of the line \(\lambda\) whose equation is \[ Lx+My+Nz=0, \] lies on the conic \(S\) whose equation is \[ (b-c)Lyz+(c-a)Mzx+(a-b)Nxy=0. \] Prove also that, if the line \(\lambda\) varies so as to pass through the point of intersection of the given lines \[ L_1x+M_1y+N_1z=0, \quad L_2x+M_2y+N_2z=0, \] then the conic \(S\) passes through a fixed point, and determine its coordinates.
H is the orthocentre and O the circumcentre of a triangle \(ABC\). \(AO\) meets the circumcircle again in \(P\). Prove that
Prove that the inverse of a circle with respect to a coplanar circle is either a circle or a straight line. \(O\) is a point in the plane of a triangle \(ABC\), and \(L, M, N\) are the feet of the perpendiculars from \(O\) to \(BC, CA, AB\). Show that by inversion with respect to \(O\) either of the two following theorems can be deduced from the other:
The foot of the perpendicular from a point \(O\) to the face \(A_2A_3A_4\) of a tetrahedron \(A_1A_2A_3A_4\) is denoted by \(A_1'\), and \(A_2', A_3', A_4'\) are similarly defined. Prove that the lines \(A_iA_j'\) and \(A_k'A_l'\) are perpendicular, where \(i, j, k, l\) are the numbers 1, 2, 3, 4 in any order. Show further that the perpendiculars from \(A_1, A_2, A_3, A_4\) to the corresponding faces of the tetrahedron \(A_1'A_2'A_3'A_4'\) are concurrent.
Find the equation of the tangent to the parabola \(y^2=4ax\) at the point \((at^2, 2at)\). A variable triangle is inscribed in the parabola \(y^2=4ax\), and two of its sides touch the parabola \(y^2=4bx\). Prove that the third side touches the parabola \(y^2=4cx\), where \[ (2a-b)^2c - ab^2 = 0. \]
Prove that four normals can be drawn to a central conic from a general point in its plane. From the point \((a\cos\theta, b\sin\theta)\) of the ellipse \(x^2/a^2+y^2/b^2-1=0\) three normals (other than the normal at the point) are drawn to the ellipse. Verify that the equation of the circle through their feet is \[ x^2+y^2 - \frac{c^2}{a}x\cos\theta - \frac{c^2}{b}y\sin\theta - a^2-b^2=0. \]
Define a homography (projectivity) between the ranges of points on two distinct lines \(l, l'\) in a plane. The points of \(l'\) corresponding under a given homography to the points \(P, Q, \dots\) of \(l\) are designated as \(P', Q', \dots\). Prove that \(PQ'\) meets \(P'Q\) on a third line \(l''\) (called the cross-axis of the homography) which depends only on the homography and is independent of the position of \(P, Q\) on \(l\). Suppose that \(l, l', l''\) are not concurrent and that a second homography is given between the ranges on \(l, l''\) whose cross-axis is \(l'\). The points of \(l''\) corresponding under this homography to \(P, Q, \dots\) on \(l\) are designated as \(P'', Q'', \dots\). Prove that the two homographies determine a composite homography between the ranges on \(l', l''\) (under which \(P', Q', \dots\) correspond to \(P'', Q'', \dots\)), whose cross-axis is \(l\).
The six coplanar points \(A, B, C, A', B', C'\) are such that \(AA', BB', CC'\) are concurrent. Prove that of the nine intersections of the sides of the triangle \(ABC\) with the sides of the triangle \(A'B'C'\) three are collinear and the remaining six lie on a conic.