\(A'\) is a variable point on the circumcircle of a given triangle \(APQ\) such that \(A\) and \(A'\) lie on the same side of \(PQ\). \(A'P, A'Q\) are produced to points \(B', C'\) respectively such that the lengths \(A'B', A'C'\) are constant and in the ratio \(\frac{AQ}{AP}\) for all positions of \(A'\). Prove that \(B'C'\) touches a fixed circle centre \(A\).
\(P\) and \(Q\) are two points lying outside a circle \(C\). Establish a method of drawing a circle through \(P\) and \(Q\), (a) to touch \(C\), (b) to intersect \(C\) orthogonally. State the number of solutions, and discuss any exceptional cases.
\(D, E, F\) are respectively the feet of the perpendiculars drawn to the sides \(BC, CA, AB\) of a triangle \(ABC\) from a point \(O\) in its plane. \(P\) is the point of intersection of the straight line through \(EF\) with the line through \(O\) parallel to \(BC\); \(Q\) that of \(DF\) with the line through \(O\) parallel to \(CA\); \(R\) that of \(DE\) with the line through \(O\) parallel to \(AB\). Prove that the points \(P, Q, R\) are collinear.
State and prove the harmonic property of the quadrangle. How many points are equidistant from four planes, no two of which are parallel? Illustrate the harmonic property of the quadrangle by considering four such points which are coplanar.
Prove that a chord of a rectangular hyperbola subtends angles at the extremities of a diameter of the hyperbola which are either equal or supplementary. Shew that a rectangular hyperbola can be drawn to pass through the orthocentre, \(H\), and the vertices \(A, B, C\) of an acute angled triangle, and having \(AH\) as a diameter. Give a geometrical construction for the asymptotes.
\(P\) is any point on a conic whose real foci are \(S, H\) and centre \(C\). Prove that the length of the chord of the circle of curvature at \(P\) which passes through \(P\) and \(C\) is \[ \frac{2SP.HP}{CP}. \] What is the corresponding result in the case of a parabola?
Obtain the condition that the pair of points given by \(ay^2+2hy+b=0\) shall be harmonic conjugates with respect to the pair given by \(a'y^2+2h'y+b'=0\). Shew that the straight lines joining any point \(P\) on a parabola to the extremities of its latus rectum are harmonic conjugates with respect to the straight line joining \(P\) to the vertex and the diameter through \(P\).
Prove that, if an ellipse is reciprocated with respect to a circle of radius \(k\) having its centre at a focus, then the reciprocal figure is a circle of radius \(\frac{k^2a}{b^2}\) whose centre is at a distance \(\frac{k^2c}{b^2}\) from that focus, where \(2a, 2b\) are respectively the lengths of the major and minor axes of the ellipse, and \(2c\) is the distance between the foci. If \(2a, 2a'\) are the lengths of the major axes of two confocal ellipses of which the first is inscribed and the second circumscribed to a triangle shew that \[ a'^4-2aa'(a'^2-c^2)-a^2c^2 = 0. \]
\(A, B, C\) are the vertices of a triangle. If points \(C', B'\) are taken in the sides \(AB, AC\) respectively such that \(\angle AC'B' = \angle ACB\), then the straight line \(C'B'\) is said to be antiparallel to \(BC\). Prove that the bisectors of the antiparallels to the sides of \(ABC\) are concurrent in a point \(K\). Shew that the antiparallels through \(K\) are diameters of a circle of radius \(\frac{abc}{a^2+b^2+c^2}\). Shew also that the area of the triangle formed by the lines through \(A, B, C\) antiparallel to the opposite sides is \[ R^2\tan A\tan B\tan C \] where \(R\) is the circumradius of the triangle \(ABC\).
Define the nine points circle of a triangle and establish the property from which it takes its name. Shew that the nine points circle touches the inscribed and escribed circles.