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.
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.