Prove that the circles which circumscribe the four triangles formed by four straight lines have a common point, and that the orthocentres of the triangles are collinear.
The points A, B, C lie on a straight line, and P is a point not on the line. The centres of the circles PBC, PCA and PAB are X, Y and Z. Prove that the four points P, X, Y, Z lie on a circle.
Prove that the equation of the normal at a point on the parabola \(x=am^2, y=2am\) is \(y+mx=2am+am^3\). \par Three normals are drawn to the parabola from the point (X, Y). Prove that if two of them are at right angles the length of the third is \(3\sqrt{(Y^2+a^2)}\).
Prove that if QOQ', ROR' are chords of a conic in fixed directions the ratio QO.OQ' : RO.OR' is constant for all positions of O. \par QOQ', ROR' are normals to an ellipse at Q, R intersecting at O at right angles and meeting the curve again at Q', R'. Prove that QQ', RR' are proportional to the distances of the centre from the tangents at Q and R.
A straight line through a fixed point P cuts a conic in A, B. Prove that the locus of the harmonic conjugate of P with regard to A, B is a straight line, the polar of P. \par C, D are conjugate points on the polar of P. Any line through P cuts the conic in A, B; AD cuts BC in E and AC cuts BD in F. Prove that E, F lie on the conic and that EF goes through P.
``Two conics are inscribed in the same triangle ABC touching BC at the same point. If from any point on BC lines are drawn touching the conics at P, Q, then PQ passes through A.'' \par State the dual theorem and prove either the theorem or its dual.
S=0, T=0, L=0 and M=0 are the equations of a conic, a tangent to the conic, a chord and a chord passing through the point of contact of T. Interpret the equations \[ S-LT=0, \quad S-MT=0, \quad S-T^2=0. \] Find the equation of the circle of curvature at the point \((x', y')\) on the conic \[ ax^2+by^2+c=0. \]
ABC is a triangle and the perpendiculars \(p,q,r\) from A, B, C to a variable straight line are such that \(p^2=\lambda qr\) where \(\lambda\) is a constant. Prove that the line envelops a conic which touches AB at B and AC at C.
Show that \(fyz+gzx+hxy=0\) is the equation in homogeneous coordinates of a conic circumscribing the triangle of reference. Find the equation of the chord joining the points \((x',y',z'), (x'',y'',z'')\) and deduce or otherwise find the equation of the tangent at \((x',y',z')\) in the form \[ fx/x'^2 + gy/y'^2 + hz/z'^2 = 0. \] Obtain the tangential equation of the curve.
Prove that, with a proper choice of a triangle of reference, the equation of a conic through four fixed points is of the form \(ax^2+by^2+cz^2=0\), where \(a,b,c\) are connected by a relation \(af^2+bg^2+ch^2=0\). \par Prove that the polars of a given point with regard to such a system of conics all pass through another point, and that if the first point lies on a given line the second lies on a conic which circumscribes the common self polar triangle of the system of conics, and find the equation of the conic.