Prove that the feet of perpendiculars from a point \(P\) of the circumcircle of the triangle \(ABC\) on to the sides lie on a straight line (the Simson Line of \(P\)). Show further that the Simson Lines of two points \(P, Q,\) of the circumcircle are inclined at an angle equal to that subtended by the chord \(PQ\) at a point of the circumference of the circumcircle, and that if they are perpendicular they intersect on the nine-point circle of triangle \(ABC\).
Prove that the inverse of a circle with respect to a coplanar circle is itself a circle or a straight line. Two circles \(C_1, C_2\) are cut by the line of centres in \(A_1, B_1\) and \(A_2, B_2\) respectively. \(P\) is any point on the radical axis and \(C\) is the circle through \(P\) orthogonal to \(C_1\) and \(C_2\). Show that by inversion with respect to a circle centre \(P\) the circles \(C_1\) and \(C_2\) can be transformed into themselves. By considering the inverse of the line of centres, or otherwise, prove that the joins of \(P\) to \(A_1\) and \(B_1\) cut \(C_1\) where it is cut by \(C\) and the joins of \(P\) to \(A_2\) and \(B_2\) cut \(C_2\) where it is cut by \(C\).
Prove that for a tetrahedron:
Define the value of the cross ratio \((ABCD)\) for four points on a straight line, and from your definition show that \((P_1H_1P_2H_2) = (Q_1H_1Q_2H_2)\) implies and is implied by \((P_1H_1Q_1H_2) = (P_2H_1Q_2H_2)\). Given that \[ (AH_1BH_2) = (BH_1CH_2) = (CH_1AH_2) = (PH_1QH_2) = (QH_1RH_2) = \lambda, \] prove that \((RH_1PH_2)\) is also \(\lambda\), where \(P, Q, R\) are three points collinear with \(A, B, C, H_1, H_2\).
Establish the harmonic property of the complete quadrangle. Prove that if a conic touches the three sides of a triangle, the joins of the vertices to the points of contact of the opposite sides are concurrent.
Prove that the two tangents which can be drawn to a parabola from the orthocentre of the triangle formed by any three tangents to the parabola are perpendicular.
Find the equation of the chord joining the two points \(P_1[ct_1, c/t_1]\) and \(P_2[ct_2, c/t_2]\) of the rectangular hyperbola \(xy = c^2\). Prove that the chord joining the two other points in which the hyperbola is cut by the circle on \(P_1P_2\) as diameter passes through the centre of the hyperbola.
By using tangential equations, or otherwise, prove that the locus of points from which perpendicular tangents can be drawn to a conic is in general a circle [Director Circle]. Show that the Director Circles of conics touching four fixed straight lines form a coaxial system.
Establish the existence of the Nine-Point circle of a triangle and prove Feuerbach's Theorem that the nine-point circle touches the inscribed and escribed circles.
Prove that the area \(A\) of a convex plane quadrilateral whose sides are of length \(a,b,c,d\) is given by \[ A^2 = (s-a)(s-b)(s-c)(s-d) - abcd \cos^2\theta, \] where \(2s=a+b+c+d\), and \(2\theta\) is the sum of one pair of opposite angles. Hence, or otherwise, show that the greatest area of a quadrilateral of given sides is obtained when it is cyclic.