\(O\) is the circumcentre, \(G\) the centroid and \(H\) the orthocentre of a triangle. Prove that \(O, G\) and \(H\) are collinear, and that \(HG=2GO\). Prove also that the nine-points centre is collinear with the other points, and that it bisects \(OH\).
Prove that the volume of a parallelepiped constructed by drawing through the opposite edges of a tetrahedron three pairs of parallel planes, is three times the volume of the tetrahedron. In a tetrahedron \(ABCD\), if \(AB\) is perpendicular to \(CD\), and \(AC\) is perpendicular to \(BD\), prove that \(AD\) is perpendicular to \(BC\).
Prove that the foot of the perpendicular from the focus on any tangent to a parabola lies on the tangent at the vertex. From a point \(O\) on the directrix of a parabola two tangents are drawn. Show that the part of the directrix intercepted by parallels through the focus to these tangents is bisected at \(O\).
Prove that the reciprocal of a circle with respect to a point \(S\) in its plane is a conic with \(S\) as focus. Prove that the envelope of chords of an ellipse which subtend a right angle at the centre is a concentric circle.
Prove that any diagonal of a complete quadrilateral is divided harmonically by its points of intersection with the other diagonals, produced if necessary. Prove, by projecting a point to infinity, or otherwise, that if \(B_0, B_2\) are harmonic conjugates to \(B_1, C\), and \(B_1, B_3\) to \(B_2, C\), and \(B_4, B_2\) to \(B_3, C\), then \(B_0, B_4\) are harmonic conjugates to \(B_2, C\).
Prove that the equation of the straight lines bisecting the angles between the lines \[ ax^2+2hxy+by^2=0 \] is \[ \frac{x^2-y^2}{a-b} = \frac{xy}{h}. \] If the distance of a given point \((p,q)\) from each of two straight lines through the origin is \(k\), prove that the equation of the straight lines is \[ (py-qx)^2=k^2(x^2+y^2). \]
Prove that the locus of the middle points of parallel chords of a parabola is a straight line parallel to the axis. Chords of the parabola \(y^2=4ax\) are drawn through the fixed point \((p,q)\). Show that the locus of their middle points is the parabola \(y(y-q)=2a(x-p)\).
Prove that the eccentric angles of ends of conjugate diameters of an ellipse differ by a right angle. The circle \(x^2+y^2+2gx+2fy+c=0\) cuts the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) in \(P, Q, R\) and \(S\). \(O\) is the centre of the ellipse. Prove that the eight extremities of diameters of the ellipse conjugate to \(OP, OQ, OR, OS\) lie, four on each, on the two circles \[ x^2+y^2 \pm \frac{2fbx}{a} \mp \frac{2gay}{b} - (a^2+b^2+c)=0, \] either upper or lower signs being taken.
Find equations to determine the foci of the conic \[ ax^2+2hxy+by^2=1. \] Find the coordinates of the foci of \[ (3x-2y)^2+4(2x+3y)^2=52, \] and sketch the curve.
If \(S=0\) is a conic, and \(L=0, M=0\) are two straight lines, interpret the equation \(S+\lambda LM=0\). A rectangular hyperbola is drawn touching the ellipse \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at \(P\), and passing through the origin. Prove that the other chord of intersection of these conics passes through a fixed point; and that for different positions of \(P\) the locus of this point is \[ a^2x^2+b^2y^2 = \frac{a^4b^4}{(a^2+b^2)^2}. \]