The inscribed circle of a triangle \(ABC\) touches the side \(BC\) at \(X\) and the inscribed circles of the triangles \(ABX, ACX\) touch \(BC\) at \(Y,Z\); prove that \(X\) bisects \(YZ\).
If \(P, Q\) are inverse points with respect to a circle \(\gamma\) and \(P', Q', \gamma'\) are the inverses of \(P, Q, \gamma\) with respect to any coplanar circle, prove that \(P', Q'\) are inverse points with respect to \(\gamma'\). Determine the locus of the inverse of a given point with respect to a variable circle of a given coaxal system, when the system has real limiting points.
A variable obtuse-angled triangle inscribed in a fixed circle with centre \(O\) has a fixed orthocentre \(H\); prove that the triangle is self-polar with respect to another fixed circle and that its sides touch a fixed conic with foci at \(O\) and \(H\).
Shew that, if there is a 1-1 correspondence between points \(P, P'\) on a straight line, there are in general two distinct points \(A, B\) on the line, such that the cross-ratio of \(A, B, P, P'\) is the same for all corresponding points \(P, P'\). State the nature of the correspondence when (i) \(A\) or \(B\) is at infinity, (ii) \(A\) and \(B\) are both at infinity.
If \(P\) is a variable point on a fixed circle and \(O\) is any point not in the plane of the circle, prove that the plane through \(P\) perpendicular to \(OP\) passes through a fixed point.
The sides of a triangle lie along the lines \(u \equiv x\cos\alpha+y\sin\alpha-p=0\), \(v \equiv x\cos\beta+y\sin\beta-q=0\), \(w \equiv x\cos\gamma+y\sin\gamma-r=0\); prove that (i) the orthocentre, (ii) the circumcentre, (iii) the centroid of the triangle are determined by the equations:
A variable line \(\lambda\) cuts the fixed conics \[ ax^2+by^2+k(a-b)=0, \quad a'x^2+b'y^2-k(a'-b')=0 \] in two pairs of points which are harmonic conjugates; prove that \(\lambda\) touches a fixed circle.
If the normals to the conic \(ax^2+by^2+c=0\) at the ends of the chord \(ahx+bky+c=0\) meet at \(P\), prove that two other normals to the conic pass through \(P\) and that the equation of the chord through the feet of these normals is \(kx+hy+hk=0\). If \(ah^2+bk^2+c=0\), deduce the coordinates of the centre of curvature at the point \((h,k)\) of the conic.
The coordinates of a variable point are \(x=(3t+1)/(t+1)\), \(y=2t/(t-1)\), where \(t\) is a parameter; prove that the locus of the point is a rectangular hyperbola and find the equation of (i) the tangent at the point with parameter \(t\), (ii) each of its asymptotes. Prove that, if the four points given by \(t=a,b,c,d\) lie on a circle, \[ a+b+c+d+bcd+cad+abd+abc=0. \]
Prove that with a proper choice of homogeneous coordinates the equation of a variable conic through four fixed points can be put into the form \(ax^2+by^2+cz^2=0\), where \(a:b:c\) are parameters and \(a+b+c=0\). Prove that two of the conics through the four fixed points touch the line \(lx+my+nz=0\), and find the coordinates of their points of contact.