Prove that \(OI^2 = R^2 - 2Rr\), where \(O, I\) are the centres of the circumscribed and inscribed circles of a triangle and \(R, r\) are the respective radii of these circles. Two equal circles \(\Gamma, \Gamma'\) with centres \(O, O'\) meet in \(P\) and the angle \(OPO'\) is \(120^\circ\). If the tangents to \(\Gamma'\) from any point on \(\Gamma\) cut \(\Gamma\) again in \(A, B\), prove that \(AB\) touches \(\Gamma'\).
Two coplanar circles meet in the points \(A, B\); \(X\) is a variable point on one circle, and \(XA, XB\) meet the second circle again in \(P, Q\). Prove that the chord \(PQ\) touches a fixed circle concentric with the second circle. Invert this theorem with regard to a circle whose centre is \(A\).
From the focus and directrix definition of a parabola, prove that the foot of the perpendicular from the focus on any tangent lies on a fixed line. Three parabolas have their foci at the vertices of a triangle and their corresponding directrices along the opposite sides. Prove that the nine common tangents (excluding the line at infinity) of the parabolas taken in pairs are concurrent in sets of three in the circumcentre, the in-centre and the three ex-centres of the triangle.
\(ABC, A'B'C'\) are two lines not in the same plane and \(AB:BC = A'B':B'C'\); prove that the lines \(AA', BB', CC'\) are parallel to a plane. Prove also that, if points \(P, Q, R\) are taken in \(AA', BB', CC'\) respectively, so that \(AP:PA' = BQ:QB' = CR:RC'\), the points \(P, Q, R\) are collinear.
Prove that the two pairs of lines \[ ax^2+2hxy+by^2=0, \quad a'x^2+2h'xy+b'y^2=0 \] are harmonically conjugate, if \(ab'+a'b-2hh'=0\). \(A, B\) are two fixed points and \(l,m\) two fixed lines in a plane, and \(P\) is a coplanar point such that \(PA, PB\) are harmonically separated by the lines through \(P\) parallel to \(l,m\). Prove that the locus of \(P\) is a hyperbola having \(AB\) as a diameter and asymptotes parallel to \(l,m\).
Two points \(P(x,y)\) and \(P'(x',y')\) in a plane are said to correspond, when their co-ordinates are connected by the relations \[ x':y':1 = 3x-y+2 : -2x+2y-2 : 2x-y+3. \] Prove that (i) the line joining any two corresponding points passes through a fixed point; (ii) the intersection of any line of points with the line of corresponding points lies on a fixed line.
Prove that the line joining the points \((r \cos \alpha, r \sin \alpha)\) and \((r' \cos\alpha, -r' \sin\alpha)\), where \(\alpha\) is fixed and \(r,r'\) are parameters satisfying the relation \(ar+br'+c=0\), touches a fixed parabola, whose directrix has equation \[ (a+b)x\cos\alpha+(b-a)y\sin\alpha+c\cos 2\alpha=0. \] Interpret this result geometrically when \(c=0\).
The tangents at the points \(T, T'\) of a conic meet in \(P\), and the tangents at the points \(U, U'\) of the conic meet in \(Q\); prove that the six points \(P, T, T', Q, U, U'\), lie on another conic. Hence, or otherwise, prove that, if \(T, T'\) are the points of contact of the tangents from a fixed point \(P\) to a variable conic of a confocal system, the circle \(PTT'\) passes through another fixed point \(Q\) and that the relation between the two points \(P, Q\) is mutual.
If \((x,y,z)\) are the homogeneous coordinates (e.g. areal or trilinear coordinates) of a point in a plane, find the equation of the polar reciprocal \(S''\) of the conic \[ S' \equiv x^2+y^2+z^2-2yz-2zx-2xy = 0 \] with respect to the conic \(S \equiv ax^2+by^2+cz^2=0\). Prove that the tangents to \(S'\) at the common points of \(S'\) and \(S''\) touch \(S\).
If \(\theta\) and \(\phi\) are unequal and less than \(2\pi\), and if \[ (x-a)\cos\theta+y\sin\theta = (x-a)\cos\phi+y\sin\phi = a, \] and \[ \tan\tfrac{1}{2}\theta - \tan\tfrac{1}{2}\phi = 2e, \] prove that \[ y^2 = 2ax - (1-e^2)x^2. \]