\(ABC\) is an acute-angled triangle in which \(AB > AC\). The internal bisector of the angle \(A\) meets \(BC\) in \(U\) and the line through \(U\) perpendicular to this bisector meets \(AB, AC\) in \(N, M\) respectively. Prove that \[ \triangle UBN > \triangle UCM. \] Determine which of the circles \(UBN, UCM\) is the larger, and prove that, if they meet again in \(D\), then \[ \frac{DB}{DC} = \frac{AB}{AC}. \]
Determine the relations between the lengths of the edges of a tetrahedron \(ABCD\) in order that a sphere may be drawn to touch its six sides, examining whether the conditions are sufficient as well as necessary.
An acute-angled triangle \(ABC\) has circumcentre \(O\) and orthocentre \(H\), and the altitude \(AH\) meets the circumcircle again in \(P\). The line \(OP\) meets \(BC\) in \(U\). Prove that an ellipse can be drawn with foci \(O, H\) to touch \(BC, CA, AB\); verify that the point of contact with \(BC\) is \(U\), and that the major axis of the ellipse is equal to the circumradius of the triangle \(ABC\).
Prove that, if \(P(x_1, y_1)\) is a point on the hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] such that \(x_1\) is positive, and if the foci are \(S(ae, 0)\), \(S'(-ae, 0)\), then \[ SP=ex_1-a, \quad S'P=ex_1+a. \] Identify the points of contact with \(SS'\) of the inscribed circle and the escribed circle opposite \(P\) of the triangle \(PSS'\).
The chord of contact of the tangents from a point \(T\) to a given ellipse meets the directrices corresponding to the foci \(S, S'\) in points \(R, R'\) respectively. Prove that \(RS, R'S'\) meet at the other end \(U\) of the diameter through \(T\) of the circle \(TSS'\). Prove that, if \(T\) varies on a fixed line \(l\), the locus of \(U\) is a hyperbola whose asymptotes are perpendicular to the major axis of the ellipse and the line \(l\).
You are given an ungraduated ruler, a pair of compasses, and a piece of paper on which are drawn two non-intersecting circles external to each other. Describe in detail, but with brevity, a construction for the radical axis and the limiting points of the circles.
The (distinct) points \(P(ap^2, 2ap)\), \(Q(aq^2, 2aq)\), \(R(ar^2, 2ar)\) of the parabola \(y^2=4ax\) are such that the focus \(S(a,0)\) is the orthocentre of the triangle \(PQR\). Prove that \[ pqr+p+q+r=0, \quad qr+rp+pq+5=0. \] Deduce, or prove otherwise, that the sides \(QR, RP, PQ\) all touch the circle \[ x^2+y^2-6ax+5a^2=0. \]
Sketch the curve \[ x^3+y^3=3xy. \] The curve is met by the rectangular hyperbola \(xy=2\) in the two (real) points \(A, B\), and \(O\) is the origin of coordinates. Prove that \[ \angle AOB = \tan^{-1}\{\sinh(\frac{1}{3}\log 2)\}. \]
A point \(P\) is taken on the diagonal \(BD\) (for convenience, produced beyond \(D\)) of the parallelogram \(ABCD\), and a line through \(P\) meets \(AD\) in \(V\) and \(AB\) in \(W\). The line through \(W\) parallel to \(AD\) meets \(CD\) in \(N\); the line through \(V\) parallel to \(AB\) meets \(BC\) in \(M\). Prove that \(M, N, P\) are collinear. The lines \(BV, DW\) meet in \(U\); the lines \(BN, DM\) meet in \(L\). Prove that \(CU\) and \(AL\) pass through the point of intersection of \(WN, VM\).
Prove that homogeneous coordinates of the points of a non-singular conic \(S\) may be expressed parametrically in the form \((\theta^2, \theta, 1)\). The polar of a point \(B\) meets the conic in \(A, C\). Prove that a conic \(\Gamma\) may be drawn through \(A, B, C\) and the vertices of the triangle of reference. The point \(B\) moves on the straight line \[ lx+my+nz=0. \] Prove that the conic \(\Gamma\) passes through the fixed point \(P(mn, -2nl, lm)\) of that line. Give a geometrical identification of \(P\) for the case when the vertices \(Z, X\) of the triangle of reference are regarded as the circular points at infinity.