Points \(D, E, F\) are taken in the sides \(BC, CA, AB\) respectively of a triangle \(ABC\). Prove that the perpendiculars to \(BC, CA, AB\) at \(D, E, F\) respectively meet in a point if and only if \[ BD^2+CE^2+AF^2 = DC^2+EA^2+FB^2. \] \(L, M, N\) are three collinear points, and \(L', M', N'\) are points in a plane through \(LMN\) such that \(LL', MM', NN'\) are perpendicular to \(LMN\). Prove that the perpendiculars drawn from \(L, M, N\) to \(M'N', N'L', L'M'\) respectively meet in a point.
\(ABC\) is a triangle, and \(X\) a point inside the triangle such that \[ \angle XBC = \tfrac{1}{3}\angle ABC, \quad \angle XCB = \tfrac{1}{3}\angle ACB. \] Points \(Y\) and \(Z\) are taken on the internal bisectors of the angles \(XBA, XCA\) respectively such that \[ \angle BXY = \tfrac{\pi}{3} + \tfrac{1}{3}\angle ACB, \quad \angle CXZ = \tfrac{\pi}{3} + \tfrac{1}{3}\angle ABC. \] Prove that the triangle \(XYZ\) is equilateral. Points \(Q, R\) are chosen respectively on \(BA\) and \(CA\) such that \(BQ=BX, CR=CX\). Prove that \(QRYZ\) are concyclic. Show that \(A\) lies on the circle through \(QRYZ\), and hence deduce that \(AY, AZ\) are the trisectors of the angle \(BAC\).
Two circles \(OAP, OAQ\) meet in \(O, A\); and \(OP, OQ\) are the diameters of the circles drawn through \(O\). The perpendiculars from \(A\) to \(OP, OQ\) meet the circles \(OAP, OAQ\) again in \(B, C\) respectively. Invert the figure with respect to \(O\), and hence prove that the circles \(BOC, POQ\) touch at \(O\).
Two circles have double contact with a parabola and touch each other. Prove that the difference between the radii of the circles is equal to the latus rectum of the parabola.
\(P\) is a variable point on an ellipse, and \(S\) is a focus. Show that the envelope of the circle on \(SP\) as diameter is the auxiliary circle of the ellipse.
Two rectangular hyperbolas meet in \(ABCD\). Show that every conic passing through \(ABCD\) is a rectangular hyperbola, and that the locus of centres of the hyperbolas is the nine-point circle of the triangle \(ABC\).
The tangential equation of a conic, referred to rectangular axes, is \[ Al^2+Bm^2+Cn^2+2Fmn+2Gnl+2Hlm = 0. \] Prove that the equation of the director circle of the conic is \[ C(x^2+y^2) - 2Gx - 2Fy + A+B=0, \] and interpret this equation when \(C=0\). Prove that the director circles of the conics inscribed in a given quadrilateral are coaxial. What is their common radical axis?
A hexagon is inscribed in a conic. Prove that the three points of intersection of pairs of opposite sides are collinear. State the dual theorem. \(A, B, C, D\) are four points on a conic; \(AC\) meets the tangent at \(D\) in \(P\) and \(BD\) meets the tangent at \(C\) in \(Q\). Prove that \(PQ\) passes through the intersection of \(AB\) and \(CD\).
\(XYZ\) is the triangle of reference and \(P\) is the point \((f,g,h)\). The line \(XP\) meets \(YZ\) in \(L\), and \(L'\) is the harmonic conjugate of \(L\) with respect to \(Y\) and \(Z\). Points \(M', N'\) are determined in a similar manner on \(YZ, ZX\) respectively. Prove that \(L'M'N'\) lie on a line, and find its equation. Show that if this line passes through the point \((p,q,r)\), then \(P\) lies on the conic \[ pyz+qzx+rxy=0. \]
The tangents to the conic \(x^2+y^2+z^2=0\) at two of its intersections with the conic \(ax^2+by^2+cz^2=0\) meet on the latter conic. Prove that the tangents to the former conic at the two remaining points of intersection also meet on the latter conic, and that the condition satisfied by \(a,b,c\) is \[ (-a+b+c)(a-b+c)(a+b-c)=0. \] (It may be assumed that \(a,b,c\) are unequal.)