Four given tangents to a circle \(C_1\) are such that through four of their mutual intersections a circle \(C_2\) can be drawn cutting \(C_1\) in real points. Prove that \(C_2\) passes through the centre of \(C_1\). Prove further, that if any four tangents to \(C_1\) are such that three of their mutual intersections lie on \(C_2\), a fourth intersection will also lie on \(C_2\).
P is a point in the plane of the triangle ABC, and L, M, and N are the feet of perpendiculars from P on to BC, CA, AB respectively. Prove that if LMN is a straight line, this line bisects the join of P to the orthocentre of triangle ABC.
A and B are points on a sphere S at opposite ends of a diameter. On the tangent plane to the sphere at B, points \(Q_1, Q_2, \dots\) etc. are taken corresponding to points \(P_1, P_2, \dots\) etc. on the sphere, where \(AP_1Q_1, AP_2Q_2, \dots\) etc. are straight lines. Prove that the curves on the plane corresponding to circles on the sphere are also circles, and that curves on the sphere and corresponding curves on the plane cut at the same angles. Show also that circles of a coaxal set on the plane are related to spheres orthogonal to the original sphere and belonging to a set with a common radical plane.
Prove that a variable straight line which is cut by the sides of a fixed triangle in segments of constant ratio touches a fixed parabola. Starting with given triangle and ratios, show how you would find the focus and directrix of the corresponding parabola by ruler and compasses construction.
Show that if the normals drawn to an ellipse at four points of it are concurrent, the conic through these four points and the centre of the ellipse is a rectangular hyperbola. Deduce that if one pair of joins of the four points is a perpendicular pair, then two of the points are at the ends of one axis of the ellipse.
Two points P, Q of a parabola are at the opposite ends of the diameter of a circle which touches the parabola at a third point. Show that the chord PQ touches another parabola having an equal and parallel latus rectum.
Prove that conics through four fixed points cut any fixed straight line in pairs of points in involution. Identify the double points of the involution. Show that for any given point P there is another point Q which is the conjugate of P with respect to every conic of the pencil. Illustrate this theorem by the special case of a set of coaxal circles.
Pairs of points \((P_r, Q_r)\) on a given straight line \(l\) are chosen so that \(AP_r, BQ_r\) intersect on a fixed straight line \(m\) which intersects \(l\), where A and B are two fixed points coplanar with \(l\) and \(m\). Prove that there are two points L and M on line \(l\) such that \((P_rL Q_r M)\) is constant.
A', B', C' are any points on the sides BC, CA, AB respectively of triangle ABC. Prove that \(AB' \cdot BC' \cdot CA' + B'C \cdot C'A \cdot A'B = 4 \cdot R \cdot \Delta'\), where R is the circumradius of ABC, and \(\Delta'\) is the area of A'B'C'. State the modification of the result which is possible in the case when AA', BB', CC' are concurrent.
Four unequal similar triangles can be drawn with sides touching a given circle of radius \(\rho\). Prove that if the areas of the triangles are \(\Delta, \Delta_1, \Delta_2\), and \(\Delta_3\), then \[ \rho^8 = \Delta \cdot \Delta_1 \cdot \Delta_2 \cdot \Delta_3. \] Prove also that the areas must satisfy the relation \[ \Delta^{\frac{1}{2}} = \Delta_1^{\frac{1}{2}} + \Delta_2^{\frac{1}{2}} + \Delta_3^{\frac{1}{2}}, \] where \(\Delta\) is the greatest of the areas, and that the angles of the triangle are \[ 2\tan^{-1}\left(\frac{\Delta_2 \cdot \Delta_3}{\Delta_1 \cdot \Delta}\right)^{\frac{1}{4}}, \] and two similar expressions.