Any point \(P\) is taken in the plane of a triangle \(ABC\). Through the mid-points of \(BC, CA, AB\) lines are drawn parallel to \(PA, PB, PC\) respectively. Prove that these lines are concurrent.
Two circles \(S, S'\) meet in \(A\) and \(B\), and the centre \(O\) of \(S\) lies on the circumference of \(S'\). From an arbitrary point \(P\) of \(S'\) tangents \(PL, PM\) are drawn to \(S\), meeting \(S'\) in \(X,Y\) respectively. \(PO\) meets \(AB\) in \(T\). Prove that
Describe the process of reciprocation with respect to a circle. Reciprocate the following theorem with respect to a circle centre \(S\), and prove either the original theorem or its reciprocal: If two conics \(\Gamma, \Gamma'\) have a common focus \(S\), and a line through \(S\) meets \(\Gamma\) and \(\Gamma'\) in \(P,Q\) and \(P',Q'\) respectively, then the tangent at \(P\) or \(Q\) meets those at \(P',Q'\) in points lying on two fixed straight lines \(a,b\) through the intersection of the corresponding directrices. Shew further that, if \(\Gamma, \Gamma'\) have the same eccentricity, \(a\) and \(b\) are at right angles.
Prove Pascal's theorem that the points of intersection of opposite sides of a hexagon inscribed in a conic are collinear. Three points of a parabola \(\Gamma\) and the direction of the axis are given. Find a geometrical construction for the point in which \(\Gamma\) meets a general line parallel to the axis.
If \(P_1, P_2, \dots, P_n\) and \(Q_1, Q_2, \dots, Q_n\) are two homographically related ranges on the same line, shew that there exist two fixed points \(E,F\) with the property \[ (EP_iFQ_i) = \text{constant (for all values of } i). \] If \(P_1, P_2, \dots, P_n\) and \(Q_1, Q_2, \dots, Q_n\) are homographically related ranges on two distinct coplanar lines \(a\) and \(b\), shew that it is possible in an infinite number of ways to find a line \(l\) and two points \(A,B\) such that the lines \(AP_i, BQ_i\) meet on \(l\) for all values of \(i\). Shew further that this is possible even if \(a\) and \(b\) coincide.
A triangle is self-polar with respect to a parabola \(\Gamma\). Prove that
Two chords \(PP', QQ'\) of a conic \(S\) are normal to \(S\) at \(P, Q\). If \(PP'\) is a bisector of the angle \(QPQ'\), prove that \(QQ'\) is a bisector of the angle \(PQP'\). If \(S\) is the parabola \[ y^2=4ax \] and \(P, Q\) are the points \((a\lambda^2, 2a\lambda), (a\mu^2, 2a\mu)\), prove that \[ \lambda\mu^3 + \mu\lambda^3 + 4\lambda\mu + 2 = 0. \]
A circle \(\Gamma\) has double contact with a hyperbola \(S\). From any point \(P\) of \(S\) a line \(PM\) is drawn parallel to an asymptote of \(S\) to meet the chord of contact in \(M\). Prove that \(PM\) is equal to the length of the tangent from \(P\) to \(\Gamma\).
The generalized homogeneous coordinates of a point of a conic \(S\) are expressed parametrically in the form \[ x:y:z = t^2:t:1. \] Shew that the sides of the triangle, whose vertices are the points whose parameters satisfy the equation \[ (at^3+bt^2+ct+d) + \lambda(a't^3+b't^2+c't+d') = 0, \] touch a fixed conic \(\Sigma\) for all values of \(\lambda\), and find the tangential equation of \(\Sigma\). Deduce that, if there exists one triangle inscribed in a given conic and circumscribed to another, then there exists an infinite number of such triangles.
Two points \(H(1,1,1)\) and \(H'(p,q,r)\) are taken in the plane of the triangle of reference \(ABC\). \(AH, AH'\) meet \(BC\) in \(L, L'\) respectively; \(M, M'\) and \(N, N'\) are similarly defined on \(CA\) and \(AB\). Prove that the six points \(L, L', M, M', N, N'\) lie on a conic \(S\), and find the equation of \(S\). \(MN, M'N'\) meet in \(P\), \(NL, N'L'\) in \(Q\), and \(LM, L'M'\) in \(R\). Prove that \(AP, BQ, CR\) are concurrent in the pole of \(HH'\) with respect to \(S\).