Prove that the ortho-centre, the centroid, the centre of the circum-circle, and the centre of the nine-points circle of a triangle lie on one straight line. Prove the property from which the nine-points circle derives its name.
Prove that any line drawn through a given point to cut a circle is divided harmonically by the circle, the point, and the polar of the point. Through a point \(A\) within a circle are drawn two chords \(PP'\) and \(QQ'\); shew that the chords \(PQ\) and \(P'Q'\) subtend equal angles at \(B\) the point conjugate to \(A\) with respect to the circle.
Define a coaxal system of circles and shew that they can be cut orthogonally by another coaxal system. Given the limiting points of a coaxal system, shew how to construct (i) the circle of the system which passes through a given point, and (ii) the circles of the system which touch a given line.
Prove that the lines joining the middle points of pairs of opposite edges of a tetrahedron are concurrent. The straight lines joining the middle points of opposite edges of a tetrahedron are equal to one another, prove that opposite edges are perpendicular to one another.
Prove that the feet of the perpendiculars from the foci on a tangent to an ellipse lie on the auxiliary circle. Given the auxiliary circle of an ellipse, a tangent, and its point of contact, construct the ellipse.
Find the condition that the line \(lx+my+n=0\) should touch the circle \(x^2+y^2+2gx+2fy+c=0\). Find the equations of the common tangents of the circles \[ x^2+y^2-2x-4y=4 \quad \text{and} \quad x^2+y^2+2x+4y=4. \]
Find the equation of the chord joining the two points on the ellipse \(x^2/a^2+y^2/b^2=1\) whose eccentric angles are \(\alpha, \beta\). A chord of an ellipse passes through a fixed point \(\xi, \eta\); prove that the locus of the middle point of the chord is \[ x^2/a^2+y^2/b^2 = x\xi/a^2+y\eta/b^2. \]
Find the directions and magnitudes of the principal axes of the conic \(ax^2+2hxy+by^2=1\). Find also the angle between the equiconjugate diameters.
Interpret the equations \(S-LM=0, S-L^2=0\); where \(S=0\) is a conic, and \(L=0, M=0\) are straight lines. Find the equation of the circle of curvature at the point on \(x^2/a^2+y^2/b^2=1\) whose eccentric angle is \(\phi\). Find also the eccentric angle of the other point in which the circle cuts the ellipse.
Show that the locus of a point such that the lengths of the tangents from it to two circles are equal, is a straight line. \(AB, CD\) are diameters of two circles and \(AC\) is parallel to \(BD\). Prove that \(AD, BC\) meet on the radical axis of the circles.