Prove that if \(D\) is the middle point of the side \(BC\) of the triangle \(ABC\), \[ AB^2+AC^2 = 2AD^2+2BD^2. \] \(PQ\) is any chord of a circle subtending a right angle at a fixed point \(O\) inside the circle whose centre is \(C\). If \(ON\) is the perpendicular from \(O\) on \(PQ\), and \(CL\) is the perpendicular from \(C\) on \(PQ\), shew that \(L\) and \(N\) lie on a fixed circle whose centre is the middle point of \(OC\).
Shew that the feet of the perpendiculars on the sides of a triangle from any point on the circumcircle lie on a straight line (the pedal line). Shew that if a chord \(PQ\) of the circumcircle of the triangle \(ABC\) is parallel to \(BC\), the pedal line of \(P\) is perpendicular to \(AQ\).
Shew that chords of a circle through a fixed point are cut harmonically by the point, its polar, and the circle. \(ABC\) is a triangle inscribed in a circle and \(O\) is the pole of \(AB\). Shew that the chord through \(O\) parallel to \(AC\) is bisected by \(BC\).
Prove that if the lines joining corresponding vertices of two coplanar triangles are concurrent, the points of intersection of corresponding sides are collinear.
Shew that the polar reciprocal of a circle with respect to a circle whose centre is \(O\) is a conic, and find the position of \(O\) if the conic is a rectangular hyperbola. Shew that chords of a fixed rectangular hyperbola which subtend a right angle at a focus touch a fixed parabola.
Shew that the locus of the middle points of parallel chords of a parabola is a straight line. A perpendicular is drawn from a fixed point \(O\) to the tangent at any point \(P\) of a parabola and cuts the diameter through \(P\) in \(Q\); prove that the locus of \(Q\) is a rectangular hyperbola, which passes through the feet of the normals from \(O\).
Shew that the locus of the intersection of perpendicular tangents to a conic is a circle (the director circle). Shew that if a focus and two tangents to a conic are given, the auxiliary circles of such conics are coaxal, and that also their director circles are coaxal.
Shew that as \(t\) varies the points given by \(\displaystyle\frac{x}{at} = \frac{b-y}{bt^2} = \frac{b+y}{b}\) lie on an ellipse. \(BB'\) is the minor axis of an ellipse and \(P\) any point on the curve. If the line through \(B\) perpendicular to \(BP\) meets \(PB'\) in \(Q\), shew that the locus of \(Q\) is a straight line.
\(PQ\) is any chord of a rectangular hyperbola and a parallel tangent touches the hyperbola at \(R\). If \(PL, QM, RN\) are the perpendiculars on an asymptote, shew that \(RN^2 = PL \cdot QM\).
Find the equation of the axes of the conic \(ax^2+2hxy+by^2+2gx+2fy+c=0\). Shew that if \(\displaystyle\frac{al+hm}{l} = \frac{hl+bm}{m} = gl+fm\), the line \(lx+my+1=0\) is a principal axis of the above conic.