From a point \(O\) on the circumcircle of a triangle \(ABC\), lines \(OL, OM, ON\) are drawn perpendicular to the sides \(BC, CA, AB\) respectively. Prove that \(L, M, N\) are collinear. Each of the lines \(OL, OM, ON\) is rotated about \(O\) through an angle \(\theta\) (in the same sense) and corresponding lines, in the new positions, meet \(BC, CA, AB\) in \(P, Q, R\) respectively. Determine whether \(P, Q, R\) are collinear.
Two given circles cut orthogonally at \(A\) and \(B\). A third circle is drawn through \(A\) to cut them in \(P\) and \(Q\) respectively, and \(O\) is a point of the third circle. Prove that the circles \(OPB, OQB\) are orthogonal.
Two circles \(S_1, S_2\) touch at a point \(C\), \(S_2\) lying inside \(S_1\). Prove that the locus of the centre of a circle touching \(S_1\) internally and \(S_2\) externally is an ellipse whose major axis is the arithmetic mean, and minor axis the geometric mean, of the diameters of the given circles.
The middle points of the edges \(AD, BC\) of a tetrahedron \(ABCD\) are \(L, M\) respectively, and \(P\) is an arbitrary point of the edge \(BD\). The line \(PL\) meets \(AB\) in \(X\), and the line \(PM\) meets \(CD\) in \(Y\). Prove that \(AX:XB = DY:YC\). Prove that \(XY\) meets \(LM\) and that, if \(U\) is the point of intersection, then each of the above ratios is equal to \(LU:UM\).
A circle is drawn passing through the foci \(S_1, S_2\) of a given ellipse and an extremity \(B\) of the minor axis. The circle meets the minor axis again in \(P\), and \(l\) is that common chord of the ellipse and the circle which is parallel to \(S_1S_2\). Prove that \(l\) is the polar of \(P\) with respect to the circle whose diameter is the minor axis of the ellipse.
The ends of the latus rectum of a parabola are \(L_1, L_2\) and \(PQ\) is a chord through the focus \(S\). Prove that \(L_1P, L_2Q\) meet in a point \(U\) of the directrix, that \(L_1Q, L_2P\) meet in a point \(V\) of the directrix, and that \(U, V\) subtend a right angle at \(S\). If \(L_1, L_2\) are replaced by two points \(L_1', L_2'\) at the ends of a focal chord other than the latus rectum, determine which, if any, of the properties corresponding to those enunciated above remain true.
The normal at a point \(P\) of a rectangular hyperbola meets the hyperbola again in \(Q\). Prove that the circle on \(PQ\) as diameter
Prove that the locus of the points of intersection of perpendicular tangents to an ellipse is a circle (the director circle). Prove that the number of rectangles circumscribed to the ellipse and inscribed in the director circle is infinite. It is proposed to draw two ellipses, intersecting in four distinct points, to have a given circle as common director circle. Determine whether, on the assumption that such ellipses do exist, their four common tangents form a rectangle, and give an example to show that it is in fact possible to find two such ellipses.
\(S\) is a given circle and \(A, B\) two given points in general position. Prove that circles through \(A\) and \(B\) cut \(S\) in pairs of points in involution. \(S\) is a given circle and \(A, B, C, D\) four given points in general position. Prove that there are two points \(X, Y\) (which may, however, be imaginary) lying on \(S\) and such that \(A, B, X, Y\) are concyclic and \(C, D, X, Y\) are concyclic. Give a geometrical construction for \(X\) and \(Y\), indicating at what stage of your construction it appears that they may be imaginary.
\(X, Y, Z, P\) are four given general coplanar points. Three conics \(S_1, S_2, S_3\) are drawn, all passing through \(P\), so that \(S_1\) touches \(XY, XZ\) at \(Y, Z\); \(S_2\) touches \(YZ, YX\) at \(Z, Y\); \(S_3\) touches \(ZX, ZY\) at \(X, Y\). Taking \(XYZ\) as triangle of reference and \(P\) as the unit point \((1,1,1)\) in general homogeneous coordinates, find the equations of the three conics and prove that they have two further common points. Describe the configuration in metrical language for the case when these two common points are the ``circular points at infinity.''