The point \(I\) is the incentre of the triangle \(ABC\). Determine under what conditions the bisector of the angle \(ABI\) meets the bisector of the angle \(ACI\) on the line \(AI\).
Two points \(A, B\) are given, and a circle is drawn such that the length of the tangent from \(A\) to it has a fixed value \(a\) and the length of the tangents from \(B\) to it has a fixed value \(b\). Find (i) the locus of the centre of this circle, and (ii) the locus, for each of the four possible points, of the point of intersection of a tangent from \(A\) to the circle with a tangent from \(B\) to the circle.
Two spheres have two distinct (real) points in common. Prove that their total intersection consists of the points of a circle.
In a tetrahedron \(OABC\) the lengths \(OA, OB, OC\) are equal and the angles \(BOC, COA, AOB\) are right angles. The foot of the perpendicular from \(O\) to the plane \(ABC\) is \(P\). Prove that the distance of \(P\) from each of the lines \(OA, OB, OC\) is the same, and equal to two-thirds of the distance of \(O\) from each of the lines \(BC, CA, AB\).
Show how to project the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \quad (a>b) \] orthogonally into a circle of radius \(b\), proving your construction. Show that it is not possible to project the curve \[ \frac{x^4}{a^4} + \frac{y^4}{b^4} = 1 \] orthogonally into a circle.
The centroid of the triangle with vertices \(P(ap^2, 2ap)\), \(Q(aq^2, 2aq)\), \(R(ar^2, 2ar)\) lies on the axis \(y=0\) of the parabola \(y^2=4ax\). Prove that the coordinates of the orthocentre (where the altitudes intersect) may be put in the form \[ x = -a(qr+rp+pq+4), \quad y = -\frac{1}{4}apqr. \] Prove that, in general, two vertices of the triangle \(PQR\) are on one side of the axis and one on the other, and that the orthocentre lies on that side of the axis where two vertices lie.
The normal at a point \(P\) in the first quadrant of the ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, \] whose foci are \(S_1(ae,0), S_2(-ae,0)\), meets the axis \(x=0\) at \(N\), and \(Q\) is the opposite end of the diameter through \(P\). Prove that a position of \(P\) exists such that the points \(S_2, Q, N\) are collinear provided that the eccentricity is greater than \(1/\sqrt{2}\).
A point \(P\) is such that two (real) perpendicular tangents can be drawn from it to the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1. \] Prove that \(a>b\) and that all such points \(P\) necessarily lie on a certain circle \(C\). The tangents from the focus \(S(ae,0)\) to the circle are parallel to the asymptotes of the hyperbola. Prove that these tangents are of length \(a\).
Prove that the circles through two fixed points \(A, B\) in a plane that cut an arbitrary line \(l\) do so in pairs of points in involution, and that this involution has (real) double points when \(A, B\) are on the same side of \(l\). Two given points \(C, D\) are in general position on the same side of \(l\) as \(A, B\). Prove that there may be two points \(P, Q\) on \(l\) with the property that \(A, B, P, Q\) are concyclic and \(C, D, P, Q\) are concyclic, establishing a condition (for example, in terms of the relative positions of the double points of the involutions determined as above by \(A, B\) and by \(C, D\)) for the existence of (real) \(P\) and \(Q\).
A point \(U\) has general homogeneous coordinates \((1,1,1)\) referred to a triangle of reference \(XYZ\). The lines \(XU, YU, ZU\) meet the sides \(YZ, ZX, XY\) in points \(P, Q, R\) respectively, and \(S\) is the conic \[ fyz+gzx+hxy=0. \] The conic through \(U, Q, R\) touching \(S\) at \(X\) meets \(YZ\) in points \(L_1, L_2\), and points \(M_1, M_2\) and \(N_1, N_2\) are defined similarly by cyclic interchange of letters. Prove that the six points \(L_1, L_2, M_1, M_2, N_1, N_2\) lie on the conic \[ \frac{x^2}{f} + \frac{y^2}{g} + \frac{z^2}{h} = 0. \]