Prove that the locus of a point in a plane whose distances from two fixed points \(L, L'\) in the plane are in a constant ratio \(\lambda\) is a circle with \(L, L'\) as inverse points. Prove that for different values of \(\lambda\) the circles form a coaxial set with \(L, L'\) as limiting points. \(\Gamma_1, \Gamma_2\) are two circles, \(\Gamma_1\) lying entirely inside \(\Gamma_2\). Their centres are \(C_1, C_2\) and their radii are \(R_1, R_2\) respectively. The length of \(C_1C_2\) is \(d\). Prove that the distance \(h\) between the limiting points of the coaxial system to which they both belong is given by \[ d^2h^2 = [(R_2+R_1)^2-d^2][(R_2-R_1)^2-d^2]. \]
From a point \(P\) in the plane of a triangle \(ABC\), the lines \(PA, PB,\) and \(PC\) are drawn to meet the opposite sides \(BC, CA,\) and \(AB\) in \(L, M,\) and \(N\) respectively. \(L', M',\) and \(N'\) are respectively the intersections of \(MN\) and \(BC\), \(NL\) and \(CA\), and \(LM\) and \(AB\). Prove that \(L', M',\) and \(N'\) lie on a straight line (the polar of \(P\) with respect to the triangle). Prove further that on the polar of \(P\) there are two and only two points \(P', P''\) whose polar lines pass through \(P\), and verify that each of \(P', P''\) lies on the polar line of the other.
\(A, B, C,\) and \(D\) are four generally placed coplanar points. \(AD\) and \(BC\) meet in \(X\), \(AC\) and \(BD\) meet in \(Y\). Prove that the midpoints of \(AB, DC,\) and \(XY\) are collinear.
Two triangles \(ABC, A'B'C'\) are inscribed in a conic \(S\). Prove that there is a conic \(\Sigma\) that can be inscribed in both triangles. Prove also that if tangents are drawn to \(\Sigma\) from any point of \(S\) to meet \(S\) again in \(Q\) and \(R\), then \(QR\) is a tangent to \(\Sigma\).
Prove that the chord of a conic \(S\) which subtends a right angle at some fixed point \(O\) in the plane of the conic will touch another fixed conic \(S'\) having \(O\) as a focus. State the particular form of this result when \(O\) is the centre of \(S\).
Determine the coordinates of the centre, and the equation and length of each principal axis, and the equation referred to these axes of the conic whose equation is \[ 17x^2 - 12xy + 8y^2 + 46x - 28y + 17 = 0. \] Find also the coordinates of the two real foci referred to the original axes.
Prove Pascal's Theorem that the intersections of opposite sides of a hexagon inscribed in a conic are collinear. Five points \(A, B, C, D,\) and \(E\) lie on a conic. If \(AB\) and \(ED\) meet in \(L\), \(CD\) and \(AE\) meet in \(M\), prove that \(BC\) and \(LM\) meet on the tangent at \(E\).
State the Eleven Point Conic Theorem in connection with the poles of a fixed straight line with respect to the conics passing through four fixed coplanar points. Prove either the theorem in general, or the theorem obtained in the special case when the fixed straight line is the ``line at infinity'' and all the conics through the four points are rectangular hyperbolas.
Given the values \(r_a, r_b,\) and \(r_c\) of the radii of the escribed circles of a triangle, find in terms of these values the lengths of the sides and the radii of the inscribed and circumscribed circles of the triangle.