\(ABC\) is a triangle, \(D\) the middle point of \(BC\); \(DG\) is drawn to cut the circle \(ABC\) in \(G\) and the angle \(CDG\) is equal to the angle \(ADC\). Prove that \(GB\) will touch the circle described about \(ABD\).
A variable line \(\lambda\) is drawn to pass through a fixed point \(O\) and meet a fixed line \(l\) in \(P\). \(Q\) is the point on \(\lambda\) conjugate to \(P\) with regard to a fixed conic \(S\). Shew that the locus of \(Q\) is a conic passing through \(O\), the pole of \(l\) with regard to \(S\), and the points of intersection of \(l\) and \(S\).
The diagram represents a roof truss composed of seven equal bars \(AE, EC, CD, DB, EF, FD, CF\) and two long horizontal bars \(AF, FB\). \(AB\) is ten feet. \emph{[A diagram of a roof truss is shown. It has vertices A, F, B on a horizontal line. Above this are vertices E, C, D forming the peak. A is below E, which is to the left of C. B is below D, which is to the right of C. F is between A and B, below C. The structure is symmetric about the vertical line through C and F. The loads are as follows: Dead weight (vertical): 1000 lb. at A, 1000 lb. at B, 2000 lb. at E, 2000 lb. at C, 2000 lb. at D. Wind pressure (perpendicular to the line CDB): 2500 lb. at B, 2500 lb. at C, 5000 lb. at D.]} The end B is fixed, the end A supported on a roller. The dead weight of the roof is distributed as indicated (1000 lb. each at A and B, 2000 lb. each at E, C, D). Wind blowing from the right causes pressures as indicated perpendicular to CDB (2500 lb. at B and C, 5000 lb. at D). Neglecting the weight of the bars, draw the force diagram. Which bars are in compression? What is the stress in the bar \(CF\)? What happens if both A and B are fixed, and what if both are on rollers?
The generalisation of metrical theorems by projection. Illustrate your account by finding the projective form of the theorem that the angle at the centre of a circle is double that at the circumference.
\(A, B, C\) are fixed points. It is required to find a point \(P\) in the plane \(ABC\) such that \(PA:PB:PC\) are given ratios. Shew how such points as \(P\) may be found by geometrical construction when the conditions can be satisfied by real points. Shew also that if \(P\) is not limited to the plane \(ABC\) its locus is a circle.
A conic \(S\) touches the sides \(BC, CA, AB\) of the triangle \(ABC\) at \(P, Q, R\) respectively. \(QR\) meets \(BC\) in \(X\). Shew that \(X\) and \(P\) are harmonically conjugate with regard to \(B\) and \(C\). Hence, or otherwise, shew that if \(P\) is the mid-point of \(BC\) the centre of \(S\) lies on \(AP\).
Of three equal discs in the same vertical plane, two rest on a horizontal table not necessarily in contact with each other, and the third rests on the first two. Shew that the least coefficient of friction between two of the discs for which this is possible is three times the least possible between a disc and the table. Can three pennies rest like the discs? The coefficient of friction between the edges of two pennies is about \(\frac{1}{4}\) and between a penny and the table about \(\frac{1}{5}\).
Shew that the system of equations \[a_{r1}x_1 + a_{r2}x_2 + a_{r3}x_3 + a_{r4}x_4 = 0 \quad (r=1, 2, 3, 4)\] can have a set of solutions, not all of which are zero, only if \[ \Delta \equiv \begin{vmatrix} a_{11} & a_{12} & a_{13} & a_{14} \\ a_{21} & a_{22} & a_{23} & a_{24} \\ a_{31} & a_{32} & a_{33} & a_{34} \\ a_{41} & a_{42} & a_{43} & a_{44} \end{vmatrix} = 0;\] and that if this condition is satisfied there is in general a simply infinite system of solutions given by \(x_r = \lambda A_r\), where \(\lambda\) is an arbitrary constant, and \(A_1, A_2, A_3, A_4\) are the constituents of any row of the reciprocal determinant, the constituents of any two such rows being proportional. In what cases is it possible to determine more than one set of values of the ratios \(x_1:x_2:x_3:x_4\) so as to satisfy the given equations?
Having given that \(\alpha, \beta, \gamma\) are the roots of the equation \[x^3 + ax^2 + bx + c = 0,\] find \(\alpha^4+\beta^4+\gamma^4\) in terms of \(a, b, c\).
Find the equation of the locus of mid-points of chords of \[\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1,\] which subtend a right angle at the point \((\alpha, 0)\).