The lines joining a point \(O\) in the plane of a triangle \(ABC\) to the vertices meet the sides \(BC, CA,\) and \(AB\) in \(D, E,\) and \(F\) respectively. If the circumcircle of the triangle \(DEF\) meets \(BC, CA,\) and \(AB\) again in \(P, Q,\) and \(R\) respectively, prove that \(AP, BQ,\) and \(CR\) are concurrent. Prove also that if the circumcircle of \(DEF\) meets \(OA\) again in \(U\), and \(OB\) again in \(V\), then \(AV\) and \(BU\) meet on \(OR\).
Prove for any tetrahedron that the perpendicular from a vertex on to the opposite face will meet the three lines drawn perpendicular to and through the orthocentres of the other three faces. Prove that if two perpendiculars from vertices on to opposite faces meet in a point \(O\), then the two perpendiculars through the orthocentres of the other two faces will also meet in \(O\). Prove that if three perpendiculars from vertices meet in \(O\), the fourth will also pass through \(O\), and the four perpendiculars at the orthocentres will also meet in \(O\). State the relationship in this last case between pairs of opposite edges of the tetrahedron.
Define inversion in plane geometry and show that orthogonal curves are inverted into orthogonal curves. The points \(P, P'\) are inverse points with respect to the circle \(C\), and the whole figure is inverted with respect to a point \(O\). Prove that \(P\) and \(P'\) are inverted into two points which are inverse with respect to the circle into which \(C\) is inverted. What happens if \(O\) lies on the circumference of \(C\)? Hence, or otherwise, prove that the centres of the circumcircles of the four triangles formed by four given straight lines are concyclic.
Prove that the locus of a point moving so that the lengths of tangents drawn from it to two fixed circles \(C_1\) and \(C_2\) are in constant ratio is a third circle \(C_3\) coaxal with \(C_1\) and \(C_2\). If for three coaxal circles \(\lambda_{12}\) denotes the fixed ratio of lengths of tangents from a point of \(C_3\) to \(C_1\) and \(C_2\) respectively, and if \(\lambda_{23}\) and \(\lambda_{31}\) are similarly defined, prove that \(\lambda_{12}.\lambda_{23}.\lambda_{31}=1\).
Find the condition for rectangular cartesian tangential coordinates that the line equation of the second degree: \[ Al^2 + Bm^2 + Cn^2 + 2Fmn + 2Gnl + 2Hlm = 0 \] should represent a parabola. Prove that the locus of points from which perpendicular tangents can be drawn to the general conic is the circle whose point equation is \[ C(x^2+y^2) - 2Gx - 2Fy + A+B=0, \] and identify the locus in the case of the parabola. Show that the circles obtained from conics touching four fixed straight lines form a coaxal set, and identify the radical axis of the set.
Prove that the mid-points of parallel chords of a conic lie on a straight line. Show that the locus of feet of perpendiculars on to parallel chords of a conic from their respective poles is, in general, a rectangular hyperbola. Identify the asymptotes of the rectangular hyperbola and show that the curve meets the original conic in points where the normal is either parallel to or perpendicular to the direction of the chords.
Show that the polar equation of a conic referred to a focus as origin may be put in the form \[ l/r=1+e\cos\theta. \] Show also that the eccentric angle \(\phi\) of a point \(P\) of an ellipse is related to the radius vector from this focus by the equation \[ r = a(1-e\cos\phi). \] Hence, or otherwise, prove for the ellipse, that \[ \tan\frac{\theta}{2} = \sqrt{\frac{1+e}{1-e}}\tan\frac{\phi}{2}. \]
Prove that if the two triangles \(ABC, PQR\) both circumscribe a conic \(\Sigma\), their vertices all lie on another conic \(S\). Prove further that if any tangent to \(\Sigma\) cut \(S\) in points \(X\) and \(Y\), the other tangents to \(\Sigma\) from \(X\) and \(Y\) meet in a point on \(S\).
Prove that if \[ \sec\alpha = \sec\beta\sec\gamma + \tan\beta\tan\gamma, \] then either \[ \begin{cases} \sec\beta = \sec\gamma\sec\alpha + \tan\gamma\tan\alpha, \\ \sec\gamma = \sec\alpha\sec\beta + \tan\alpha\tan\beta; \end{cases} \] or \[ \begin{cases} \sec\beta = \sec\gamma\sec\alpha - \tan\gamma\tan\alpha, \\ \sec\gamma = \sec\alpha\sec\beta - \tan\alpha\tan\beta. \end{cases} \]
If the triangle \(ABC\) has sides of length \(a,b\), and \(c\), respectively, and if with the usual notation \(r, r_1, r_2\) and \(r_3\) denote the radii of the inscribed and escribed circles respectively, prove that \[ r_2 r_3 \tan\frac{1}{2}A \quad \text{and} \quad \frac{r_2-r_3}{b-c}, \] are equal, and determine their common value in terms of \(a,b\), and \(c\).