If \(H\) is the orthocentre of a triangle \(ABC\) and if \(AH\) cuts \(BC\) in \(D\) and the circumcircle again in \(X\), prove that \(HD=DX\), with similar results for \(BH\) and \(CH\). Derive from this result another theorem by inverting with centre \(H\).
Prove that, if a variable chord of a circle subtends a right angle at a fixed point, the locus of its pole is a circle.
Two points \(A, B\) in space are on the same side of a plane. Find a point \(P\) in the plane such that the sum of the distances \(PA, PB\) is a minimum.
The base \(BC\) of a triangle is given. Find the locus of the vertex \(A\) when (i) the sum of the base angles \(B, C\) is given; (ii) the difference of the base angles is given.
Two rectangular hyperbolas intersect in \(A, B, C, D\). Prove that all conics through \(A, B, C, D\) are rectangular hyperbolas. \par Deduce that (i) the orthocentre of a triangle inscribed in a rectangular hyperbola lies on the curve; (ii) if a chord of rectangular hyperbola subtends a right angle at a point \(O\) of the curve, then the chord is parallel to the normal at \(O\).
\(A, B, C, D\) are four collinear points whose cross-ratio \((ABCD)\) is \(-\tan^2\theta\). Find, in terms of \(\theta\), the cross-ratios \((ADCB), (ACBD), (ADBC), (ABDC), (ACDB)\). \par Four points \(P_1, P_2, P_3, P_4\), collinear with \(A, B, C\), are such that \((ABCP_r) = -\tan^2\theta_r\), \(r=1,2,3,4\). Prove that \[ (P_1 P_2 P_3 P_4) = \frac{\sin(\theta_2-\theta_1)\sin(\theta_4-\theta_3)}{\sin(\theta_4-\theta_1)\sin(\theta_2-\theta_3)} \frac{\sin(\theta_2+\theta_1)\sin(\theta_4+\theta_3)}{\sin(\theta_4+\theta_1)\sin(\theta_2+\theta_3)}. \]
If the sides of two triangles touch a conic, prove that their vertices all lie on a conic. \par State the relation between the foci of a conic and the circular points and deduce that, if a parabola touches each of three given straight lines, then its focus lies on the circumcircle of the triangle formed by these lines.
Prove that the circles described on the three diagonals of a complete quadrilateral are coaxal.
Express \[ \begin{vmatrix} \sin^3\theta & \sin\theta & \cos\theta \\ \sin^3\alpha & \sin\alpha & \cos\alpha \\ \sin^3\beta & \sin\beta & \cos\beta \end{vmatrix} \] as the product of four sines and hence find all values of \(\theta\), in terms of \(\alpha\) and \(\beta\), for which the value of this determinant is zero.
If \(O, H, I, K\) are respectively the centres of the circum-, ortho-, in-, and nine-point-circles of a triangle \(ABC\), prove that \(IH^2 = 2r^2-4R^2\cos A\cos B\cos C\), and obtain similar expressions for \(OI^2\) and \(OH^2\), where \(r, R\) are the radii of the in- and circum-circles. \par Deduce that \(IK = \frac{1}{2}R-r\), and hence that the in- and nine-point-circles touch.