Squares \(BCLP, CAMQ, ABNR\), of centres \(X, Y, Z\), are described outwards on the sides \(BC, CA, AB\) of a triangle \(ABC\). The circles \(CAMQ, ABNR\) meet in \(O\). Prove that \(O\) lies on each of the lines \(BM, CR, NQ, AX\) and on the circle \(BCX\). Prove also that \(AX\) is equal to \(YZ\).
A point \(U\) is taken on the circumcircle of a triangle \(ABC\), and \(P, Q, R\) are the feet of the perpendiculars from \(U\) to \(BC, CA, AB\). Prove that \(P, Q, R\) lie on a straight line (the Simson or pedal line of \(U\) with respect to the triangle \(ABC\)). The altitudes of the triangle \(ABC\) meet the circumcircle again in \(L, M, N\). Prove that the triangle formed by the Simson lines of \(L, M, N\) with respect to the triangle \(ABC\) has its sides parallel and equal to those of the triangle \(LMN\).
Two lines \(l, p\) meet in a point \(U\). Points \(L, M, N\) are taken on \(l\) and points \(P, Q, R\) are taken on \(p\); these points are all distinct, and different from \(U\). Prove that the points of intersection \((MR, NQ), (NP, LR), (LQ, MP)\) lie on a line \(x\). Prove that, if \(LP, MQ, NR\) meet in a point, then \(x\) passes through \(U\). If \(x\) passes through the point \(U\), determine whether the lines \(LP, MQ, NR\) are necessarily concurrent.
Two points \(P, Q\) lie inside a sphere of radius \(a\) and centre \(O\), and \(OP=p, OQ=q, \angle POQ=\theta\). The planes through \(P, Q\), perpendicular to \(OP, OQ\) respectively, intersect in a line \(l\). Prove that \(l\) cuts the sphere if \[ p^2+q^2-2pq\cos\theta < a^2\sin^2\theta, \] and find what length of the line \(l\) then lies inside the sphere.
The point \(P\) on the ellipse \(b^2x^2+a^2y^2=a^2b^2\) with foci \(S, S'\) has coordinates \((a\cos\theta, b\sin\theta)\). Find the coordinates of the centre of the escribed circle opposite to \(P\) of the triangle \(PSS'\), and the locus of this centre as \(P\) moves on the ellipse.
The normal at the point \(P(ap^2, 2ap)\) of the parabola \(y^2=4ax\) meets the parabola again at the point \(N(an^2, 2an)\). The line joining \(P\) to the focus \(S(a,0)\) meets the parabola again in \(R(ar^2, 2ar)\). Prove that the line \(NR\) meets the axis \(y=0\) on the side of the directrix \(x+a=0\) remote from the parabola. Prove also that if \(Q(aq^2, 2aq)\) is another point of the parabola such that the normal at \(Q\) passes through \(N\), then, for varying positions of \(P\), the line \(QR\) envelopes the parabola whose equation is \(y^2+32ax=0\).
A variable circle through the foci \((\pm ae, 0)\) of the hyperbola \(b^2x^2-a^2y^2=a^2b^2\), where \(b^2=a^2(e^2-1)\), meets the hyperbola in four points \(P, Q, R, S\), named so that \(PQ\) is parallel to \(RS\). The lines \(PQ, RS\) meet the axis \(x=0\) in \(U, V\). Prove that \(OU \cdot OV\) is constant for all the circles of the system, where \(O\) is the origin of coordinates. Prove also that \(UV=2(1-e^{-2})r\), where \(r\) is the radius of the circle \(PQRS\).
It is required to determine whether there is a point \(P(x_1, y_1)\) on the ellipse \[ b^2x^2+a^2y^2 = a^2b^2, \] of centre \(O\) and foci \(S(ae,0), S'(-ae,0)\), with the property that the circle \(POS\) touches the ellipse at \(P\). Prove that, if there is such a point, its abscissa \(x_1\) satisfies the equation \[ (ex_1-a)(ex_1^2+ax_1-ea^2) = 0, \] and show that there is one, and only one, relevant value of \(x_1\).
A triangle \(ABC\) is inscribed in a conic \(S\), and \(O\) is a point on the conic. A straight line through \(O\) meets \(BC, CA, AB\) in \(L, M, N\), and the conic again in \(U\). Prove that the cross-ratio \((LMNU)\) on the line is equal to the cross-ratio \((ABCO)\) on the conic. Two triangles \(ABC, OUV\) are inscribed in a conic \(S\). Prove that their six sides touch a conic \(\Sigma\).
Prove that the coordinates of a general point of a conic may be expressed in terms of suitably chosen homogeneous coordinates in the parametric form \((t^2, t, 1)\). The tangents at \(Z, X\) to a conic meet in \(Y\), and the tangents at \(C, A\) meet in \(B\). Prove that the six points \(X, Y, Z, A, B, C\) lie on a conic and that the tangents at \(Y, B\) to that conic pass through the point of intersection of \(ZX\) and \(CA\).