Prove that the reciprocal of a circle with respect to another circle whose centre is \(S\) is a conic with \(S\) as focus. Two conics have a common focus \(S\) and two real common tangents; from a variable point \(P\) on one of the common tangents two other tangents \(PX, PY\) are drawn to meet the second common tangent in \(X\) and \(Y\); prove that the angle \(XSY\) is constant.
The origin of a pair of rectangular axes in a plane is transferred to the point \(a, b\), and the axes are turned through an angle \(\theta\); show that there is one point which has the same coordinates referred to either set of axes; find the coordinates. Show also that the point forms with the two origins of coordinates an isosceles triangle of vertical angle \(\theta\).
Show that there is in general one circle of a coaxal system which cuts a given circle orthogonally. Prove also that, if the distance of the centre of any circle of the system of radius \(r\) from the radical axis is \(\kappa\) and the angle at which it cuts the given circle is \(\theta\), \(\frac{\kappa-\kappa_0}{r}\sec\theta\) is constant for the system, where \(\kappa_0\) is the value of \(\kappa\) for the circle which cuts orthogonally.
Show that four normals can be drawn from a given point to the conic \(ax^2+by^2=1\), and show that if the chord joining two of the feet of the normals is \(lx+my=1\), the chord joining the other two feet is \(\frac{ax}{l}+\frac{by}{m}=-1\). Prove that the normals to the conic at its intersection with \(lx+my=1\) meet at the point \[ \left( \frac{l(a-b)(m^2-b)}{b(am^2+bl^2)}, \frac{m(b-a)(l^2-a)}{a(am^2+bl^2)} \right). \]
Show that the locus of the pole of a given line with respect to a series of confocal conics is a straight line. The normal at \(P\) to \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\) meets the polar of \(P\) with respect to the confocal \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=a+b\) in \(Q\). Show that \(PQ\) is equal to the semi-diameter of the first conic parallel to the tangent at \(P\), and that \(Q\) lies on the circle \(x^2+y^2=(a+b)^2\).
A rectangular hyperbola is cut by any circle in four points. Prove that the sum of the squares of the distances of the points from the centre of the hyperbola is equal to the square of the diameter of the circle.
Find the tangential equation of the conic \[ ax^2+by^2+cz^2+2fyz+2gzx+2hxy=0, \] and determine the tangents common to \[ x^2+y^2+4zx-2xy=0, \quad 16x^2-3y^2+5z^2-2yz=0, \] together with the tangential equation of one of the points of contact with the first of these conics.
Points \(X, Y, Z\) are taken in the sides \(BC, CA, AB\) of an equilateral triangle \(ABC\) and \(AX, BY, CZ\) form the triangle \(PQR\). Prove that, if the triangle \(PQR\) is equilateral, so also is the triangle \(XYZ\).
Prove that the common tangents to two circles whose centres are \(A\) and \(B\) cut the line \(AB\) in the points which divide \(AB\) internally and externally in the ratio of the radii of the circles. Prove also that the other points in which the common tangents intersect each other lie on the circle whose diameter is \(AB\).
The tangents to the circumcircle of a triangle \(ABC\) cut the opposite sides in \(X, Y, Z\). Prove the circles whose diameters are \(AX, BY, CZ\) have a common radical axis which passes through the orthocentre and the circumcentre of the triangle.