Prove that, if \(P, A, B, C\) are four points in a plane, there is another point \(P'\) in the plane such that \[ P'A:P'B:P'C = PA:PB:PC, \] and that \(P'\) coincides with \(P\), if and only if \(P\) lies on the circle \(ABC\).
Prove that the inverse of a straight line is a circle through the centre of inversion. Circles \(BDC, CEA, AFB\) cut the circle \(ABC\) orthogonally. Through \(A, B, C\) circles \(AD, BE, CF\) are drawn orthogonal to \(ABC\) and to \(BDC, CEA, AFB\) respectively. Prove that the circles \(AD, BE, CF\) meet in two points.
Prove that the polar reciprocal of one circle with regard to another is a conic. A conic is drawn with a focus at the centre \(C\) of a given circle so as to touch the circle and pass through a fixed point \(O\) on the circumference. Prove that the corresponding directrix touches the circle on \(CO\) as diameter.
\(A, B, C, D\) are the common points of two conics \(S, S'\). Prove, by projection or otherwise, that if the tangents to \(S\) at \(A, B\) meet on \(S'\), then the tangents to \(S\) at \(C, D\) also meet on \(S'\). State the dual theorem.
Prove that a chord of an ellipse which subtends a right angle at a given point \(P\) on the curve cuts the normal at \(P\) in a fixed point \(Q\). Prove that if the equation of the ellipse be \(x^2/a^2+y^2/b^2=1\), the locus of \(Q\) corresponding to different positions of \(P\) is \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = \frac{(a^2-b^2)^2}{(a^2+b^2)^2}. \]
Find the conditions that the lines \(lx+my+n=0\), \(l'x+m'y+n'=0\) may be conjugate diameters of the conic \[ ax^2+2hxy+by^2+2gx+2fy+c=0 \quad (ab \ne h^2). \] Prove that the equation of the pair of lines which are conjugate diameters of both the conics \(ax^2+2hxy+by^2=1\) and \(a'x^2+2h'xy+b'y^2=1\) is \[ (ah'-a'h)x^2+(ab'-a'b)xy+(hb'-h'b)y^2=0. \]
Prove that, if \(A\) is any point on a conic and \(PQR\) is a self-conjugate triangle and \(AQ, AR\) meet the conic in \(B, C\), then \(P, B, C\) are collinear. Prove that, if \(ABC\) is a triangle inscribed in a conic, then there is an infinite number of self-conjugate triangles \(PQR\) such that \(P\) lies on \(BC\), \(Q\) on \(CA\) and \(R\) on \(AB\). Prove that \(AP, BQ, CR\) intersect on the conic.
Find the coordinates of the eight points of contact of the common tangents of the conics \(x^2+y^2+z^2=0\), \(ax^2+by^2+cz^2=0\), and shew that they lie on the conic whose equation is \[ a(b+c)x^2+b(c+a)y^2+c(a+b)z^2=0. \]
A conic circumscribes a triangle \(ABC\) and its centre lies on the median through \(A\). Prove that its asymptotes envelop a conic which touches \(AB\) and \(AC\) at \(B\) and \(C\).
Prove that the conics given by the equations \(z^2+2hxy=0\), \(z^2+2hxy+2fyz=0\) have three-point contact. Prove that, if the conics touch at \(A\) and cut at \(B\), and \(P,Q\) are the points of contact of the other common tangent, then the tangent at \(A\) is the harmonic conjugate of \(AB\) with regard to \(AP\) and \(AQ\).