Problems

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.

\(P\) is any point in the plane of a triangle \(ABC\), and \(X\) is the reflexion of \(P\) in the side \(BC\) (i.e. \(BC\) is the perpendicular bisector of \(PX\)); \(Y, Z\) are the reflexions of \(P\) in the sides \(CA, AB\) respectively. Prove that the circles \(BCX, CAY, ABZ\) have a common point \(Q\), and that in general the relation between the points \(P, Q\) is mutual. Discuss the cases in which (i) \(P\) is on the circumcircle \(ABC\), (ii) \(P\) is the orthocentre of the triangle \(ABC\).

A variable line through a fixed point \(O\) cuts a fixed conic in points \(P, Q\); \(X\) is the harmonic conjugate of \(O\) with respect to \(P, Q\). Prove that the locus of \(X\) is a straight line. The tangents to a conic at the points \(A, B\) of the conic meet on the normal at a point \(C\) of the conic; prove that the chords \(CA, CB\) are equally inclined to the normal at \(C\).

If \(A, B, C, D\) are any four coplanar points, prove that the three pairs of lines through any point parallel to \(BC\) and \(AD\), \(CA\) and \(BD\), \(AB\) and \(CD\) are in involution. If \(A, B, C\) are fixed, find the locus of \(D\) when the double lines of this involution are perpendicular to each other.