If \(0< \theta < \alpha < \phi < 2\pi\) and \(\alpha+\beta=\theta+\phi<2\pi\), show that \[ \sin\alpha + \sin\beta > \sin\theta + \sin\phi. \] Prove that among the \(n\)-sided polygons inscribed in a given circle, the regular ones (those whose sides are all equal) enclose the greatest area.
If \(l,m,p\) and \(q\) are real numbers and \(lm<0\), show that the equations \[ xy=p, \quad (y-lx)(y-mx)=q \] have a real solution. By taking \(\beta x - \alpha y\) and \(\beta x + \alpha y\) as new variables, show that if \(\alpha, \beta, \gamma\) are the sides of a triangle, the equations \[ \frac{a^2+x^2}{\alpha^2} = \frac{b^2+y^2}{\beta^2} = \frac{c^2+(x-y)^2}{\gamma^2} \] always have a real solution. (\(a,b,c\) are real.) Deduce that, given any two triangles, the first may always be regarded as the orthogonal projection of a triangle similar to the second.
The point \(O\) is the centre of the circle \(PQR\) and the tangents at \(O\) to the circles \(OQR\) and \(OPR\) meet the circles \(OPR\) and \(OQR\) again in \(X\) and \(Y\), respectively. Show that, if \(PX\) and \(QY\) meet the circle \(PQR\) again in \(U\) and \(V\), then the parabola which touches \(OX\) and \(OY\) at \(X\) and \(Y\) has focus \(R\) and directrix \(UV\).
Find the coordinates of the centre, the equations of the axes and the lengths of the semi-axes of the ellipse \[ 41x^2 - 24xy + 34y^2 + 106x - 92y + 49 = 0. \]
Show that, as \(t\) varies, the point \[ x = \frac{a_1t^2+2h_1t+b_1}{a_3t^2+2h_3t+b_3}, \quad y = \frac{a_2t^2+2h_2t+b_2}{a_3t^2+2h_3t+b_3} \] in general describes a conic \(S\), and find the condition that the line \(lx+my+n=0\) shall touch \(S\). Prove that the director circle of \(S\) is \[ (a_1-xa_3)(b_1-xb_3) + (a_2-ya_3)(b_2-yb_3) = (h_1-xh_3)^2 + (h_2-yh_3)^2. \]
If \(ABCD\) is a tetrahedron, prove that the lines joining the vertices \(A,B,C,D\) to the centroids of the opposite faces are concurrent. Show further that, if \(AB\) is perpendicular to \(CD\) and \(AC\) is perpendicular to \(BD\), then \(AD\) will be perpendicular to \(BC\).
P, Q, R are three collinear points, and O is a point not on the line PQR. Lines are drawn through P, Q, R perpendicular to OP, OQ, OR respectively, to form a triangle ABC in which P is on BC, Q on CA and R on AB. Prove that
The tangents from a point P to two non-intersecting coplanar circles are equal. Prove that the locus of P is a straight line (the radical axis of the two circles). The incircle of a triangle ABC touches the sides BC, CA, AB at D, E, F respectively, and the escribed circle opposite A touches them at P, Q, R respectively. The middle points of EQ, FR are Y, Z. Prove that the straight line YZ bisects BC.
Given a parallelogram ABCD, establish the existence of an ellipse touching each of the four sides at its middle point. Hence, or otherwise, prove that a parallelogram may be projected into a square by a single orthogonal projection.
ABCD is a given tetrahedron. A circle in the plane ABC meets BC, CA, AB in the pairs of points \(P_1, P_2\); \(Q_1, Q_2\); \(R_1, R_2\) respectively. A circle in the plane DBC passes through \(P_1, P_2\) and meets DB, DC in the pairs of points \(M_1, M_2\); \(N_1, N_2\) respectively. Prove that \(Q_1, Q_2, N_1, N_2\) are concyclic and that \(R_1, R_2, M_1, M_2\) are concyclic, and that these two circles meet in two points \(L_1, L_2\) on AD. The lines \(M_1N_2, M_2N_1\) meet at \(X_1\), and \(M_1N_1, M_2N_2\) at \(X_2\). The lines \(N_1L_2, N_2L_1\) meet at \(Y_1\), and \(N_1L_1, N_2L_2\) at \(Y_2\). The lines \(L_1M_2, L_2M_1\) meet at \(Z_1\), and \(L_1M_1, L_2M_2\) at \(Z_2\). Prove that the six points \(X_1, X_2, Y_1, Y_2, Z_1, Z_2\) are coplanar.