If \(ABC\) is an acute-angled triangle, show how to construct the point \(P\) at which all the sides subtend 120\(^\circ\). \newline \(AP\) cuts the circle \(PBC\) in \(L\); \(BP\) cuts the circle \(PCA\) in \(M\); \(CP\) cuts the circle \(PAB\) in \(N\). Prove that \(AL=BM=CN\).
\(A_1, A_2, A_3, A_4\) are the vertices of a quadrangle; \(G_1\) is the centroid of \(A_2A_3A_4\); \(G_2, G_3, G_4\) are similarly defined. Prove that \(G_1G_2G_3G_4\) is a quadrangle similar to \(A_1A_2A_3A_4\). \newline If \(A_1, A_2, A_3, A_4\) are concyclic, and if \(H_1\) is the orthocentre of \(A_2A_3A_4\) and \(H_2, H_3, H_4\) are similarly defined, prove that \(H_1H_2H_3H_4\) is a quadrangle similar to \(A_1A_2A_3A_4\).
Define the cross-ratio of four points on a circle. Prove this to be unchanged by inversion. \newline \(A, B\) are fixed points and \(\Sigma\) is a fixed circle; \(B'\) is the inverse of \(B\) with respect to \(\Sigma\). A variable circle through \(A, B\) cuts \(\Sigma\) in \(X\) and \(Y\); \(P\) is the harmonic conjugate of \(B\) with respect to \(X, Y\) on the variable circle. Prove that the locus of \(P\) is a circle through \(A\) and \(B'\) and orthogonal to the circle \(ABB'\).
\(A_1, A_2, B_1, B_2\) are four points in space. \(C_1\) divides \(A_1B_1\) in the ratio \(\lambda:1\) and \(C_2\) divides \(A_2B_2\) in the same ratio; \(A_3\) divides \(A_1A_2\) in the ratio \(\mu:1\) and \(B_3\) divides \(B_1B_2\) in the same ratio. Prove that \(A_3B_3\) meets \(C_1C_2\), and that their point of intersection divides \(A_3B_3\) in the ratio \(\lambda:1\) and \(C_1C_2\) in the ratio \(\mu:1\).
Between the points of two lines, \(l\) and \(l'\), of the plane a homography is given which pairs \(A, B, C, \dots\) of \(l\) with \(A', B', C', \dots\) of \(l'\). Prove that \(BC'\) and \(B'C\), \(CA'\) and \(C'A\), \(AB'\) and \(A'B, \dots\) meet on a line, \(\lambda\), called the cross-axis of the homography. \newline If a different homography between \(l\) and \(l'\) pairs \(A, B, C, \dots\) with \(A'', B'', C'', \dots\), prove that \(P'=P''\) for at most two points \(P\) of \(l\). \newline Two different homographies between \(l\) and \(l'\) both pair a given point \(A\) of \(l\) with \(A'\) of \(l'\). Their cross-axes, \(\lambda\) and \(\mu\), are given. Find a construction which will give the other point, \(B\), of \(l\) whose pairs under the two homographies are the same.
Prove that chords of the ellipse, \(S\), with equation \(\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\), which subtend a right angle at the point \((a \cos \alpha, b \sin \alpha)\) all pass through a fixed point, called its Frégier point, and find its co-ordinates. \newline Prove that the locus of all Frégier points as \(\alpha\) varies is an ellipse \(S'\), similar to \(S\) and similarly situated. Prove that the circle on a chord of \(S\) as diameter will touch \(S'\) if the chord touches \(S'\).
\(XYZ\) is a triangle in a projective plane and \(P\) is a coplanar point. \(XP\) cuts \(YZ\) in \(L\), and \(M, N\) are similarly defined; \(MN\) cuts \(YZ\) in \(L'\) and \(M', N'\) are similarly defined. Prove that \(L', M', N'\) lie on a line, called the polar of \(P\) with respect to \(XYZ\). \newline If \(P\) varies so that its polar passes through a point \(Q\), prove that \(P\) traces out a conic through \(X, Y, Z\). If \(Q\) varies on a line, \(l\), prove that its associated conic always passes through a point, \(R\), other than \(X, Y, Z\), and that \(l\) is the polar of \(R\).
Tangents \(PA, PB, QC, QD\) are drawn to a conic from two points \(P\) and \(Q\), the points of contact being \(A, B, C, D\). Prove that \(PQ\) is a side of the diagonal point triangle of the quadrangle \(ABCD\), and that the six points \(A, B, C, D, P, Q\) lie on a conic.
\(A, B, C\) are three points, \(AT\) and \(AU\) two lines through \(A\) not containing \(B\) or \(C\). A variable conic through \(A, B, C\) and touching \(AT\) at \(A\) cuts \(AU\) in \(P\); the pole of \(BC\) with respect to this conic is \(Q\). Prove that \(PQ\) cuts \(BC\) in a fixed point \(K\). \newline If now \(AT\) varies in direction, prove that \(K\) varies homographically on \(BC\).
A conic \(S\) is inscribed in the triangle \(ABC\) and \(K\) is a point of \(S\). A variable point \(P\) is taken on the tangent at \(K\) and \(PT\) is the other tangent from \(P\) to \(S\). Prove that \(P\{ABCT\}\) is constant. \newline Reciprocate with respect to a conic whose centre is \(A\) and state the theorem which can be deduced.