P, Q, R are three collinear points, and O is a point not on the line PQR. Lines are drawn through P, Q, R perpendicular to OP, OQ, OR respectively, to form a triangle ABC in which P is on BC, Q on CA and R on AB. Prove that
The tangents from a point P to two non-intersecting coplanar circles are equal. Prove that the locus of P is a straight line (the radical axis of the two circles). The incircle of a triangle ABC touches the sides BC, CA, AB at D, E, F respectively, and the escribed circle opposite A touches them at P, Q, R respectively. The middle points of EQ, FR are Y, Z. Prove that the straight line YZ bisects BC.
Given a parallelogram ABCD, establish the existence of an ellipse touching each of the four sides at its middle point. Hence, or otherwise, prove that a parallelogram may be projected into a square by a single orthogonal projection.
ABCD is a given tetrahedron. A circle in the plane ABC meets BC, CA, AB in the pairs of points \(P_1, P_2\); \(Q_1, Q_2\); \(R_1, R_2\) respectively. A circle in the plane DBC passes through \(P_1, P_2\) and meets DB, DC in the pairs of points \(M_1, M_2\); \(N_1, N_2\) respectively. Prove that \(Q_1, Q_2, N_1, N_2\) are concyclic and that \(R_1, R_2, M_1, M_2\) are concyclic, and that these two circles meet in two points \(L_1, L_2\) on AD. The lines \(M_1N_2, M_2N_1\) meet at \(X_1\), and \(M_1N_1, M_2N_2\) at \(X_2\). The lines \(N_1L_2, N_2L_1\) meet at \(Y_1\), and \(N_1L_1, N_2L_2\) at \(Y_2\). The lines \(L_1M_2, L_2M_1\) meet at \(Z_1\), and \(L_1M_1, L_2M_2\) at \(Z_2\). Prove that the six points \(X_1, X_2, Y_1, Y_2, Z_1, Z_2\) are coplanar.
A'A is the major axis of an ellipse of centre O and foci S', S. The tangent at a point P of the ellipse meets the major axis in T, the normal at P meets the major axis in G, and the perpendicular from P (the ordinate through P) meets it in N. Prove that \[ ON.OT=OA^2, \quad OG.OT=OS^2. \] Deduce that the eccentricity of the ellipse is \(\sqrt{(OG/ON)}\), and that the length of the minor semi-axis is \(\sqrt{(NG.OT)}\).
The tangents to the ellipse \(\displaystyle\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) at the points \(L_1, M_1\) meet at \(P_1\), and the tangents at \(L_2, M_2\) meet at \(P_2\). Prove that the six points \(L_1, M_1, P_1, L_2, M_2, P_2\) lie on a conic. Prove also that, if this conic is a rectangular hyperbola, then the points \(P_1, P_2\) are conjugate with respect to the director circle of the ellipse.
The normal to the parabola \(y^2=4ax\) at the point \(P(ap^2, 2ap)\) meets the curve again in \(N(an^2, 2an)\), and the line joining \(N\) to the focus \((a,0)\) meets the curve again in \(Q(aq^2, 2aq)\). Prove that \[ q = p/(p^2+2). \] Deduce that there are two positions of \(P\), say \(P_1\) and \(P_2\), which give rise in this way to the same point \(Q\), and prove that all the straight lines joining corresponding points \(P_1, P_2\) meet the axis of the parabola in a fixed point.
S is a fixed point outside a given circle of centre S'. An arbitrary point U is taken on the circle, and the perpendicular bisector of SU meets the line through S' and U in a point P. Prove that the locus of P is a hyperbola. Identify the two parts of the circle on which U must lie in order to give rise to the two branches of the hyperbola, and determine the four positions of U on the circle for which the corresponding point is at an end of a latus rectum.
Prove that the equation of the parabola which touches the rectangular hyperbola \(xy=c^2\) at each of the points U(\(cu, c/u\)), V(\(cv, c/v\)) is \[ x^2 - 2uvxy + u^2v^2y^2 - 2c(u+v)x - 2cuv(u+v)y + c^2(u^2+6uv+v^2)=0. \] The axis of this parabola cuts the principal axes of the hyperbola (\(x\pm y=0\)) in P and Q, and also cuts the line UV in R. Prove that R is the middle point of PQ and also that R is the foot of the perpendicular from the origin to UV.
Obtain the equation of a conic inscribed in the triangle of reference XYZ of general homogeneous coordinates in the form \[ x^2+y^2+z^2-2yz-2zx-2xy=0. \] The line joining X to the point of contact on YZ cuts the conic again in U; points V, W are defined similarly by cyclic interchange of the letters X, Y, Z. Prove that the conic UVWYZ touches the given conic at U.