1. The condition that a triangle \((xyz)\) should be equilateral is that \[x^{2} + y^{2} + z^{2} – yz – zx – xy = 0.\]
[Let $XYZ$ be the triangle. The displacement $\overline{ZX}$ is $\overline{YZ}$ turned throughan angle~$\frac{2}{3}\pi$ in the positive or negative direction. Since $\operatorname{Cis}\frac{2}{3}\pi = \omega_{3}$,$\operatorname{Cis}(-\frac{2}{3}\pi) = 1/\omega_{3} = \omega_{3}^{2}$, we have $x – z = (z – y)\omega_{3}$ or $x – z = (z – y)\omega_{3}^{2}$. Hence $x + y\omega_{3} + z\omega_{3}^{2} = 0$ or $x + y\omega_{3}^{2} + z\omega_{3} = 0$. The result follows from Exs. XXII. 3.]
2. If \(XYZ\), \(X’Y’Z’\) are two triangles, and \[\overline{YZ} \cdot \overline{Y’Z’} = \overline{ZX} \cdot \overline{Z’X’} = \overline{XY} \cdot \overline{X’Y’},\] then both triangles are equilateral. [From the equations \[(y – z)(y’ – z’) = (z – x)(z’ – x’) = (x – y)(x’ – y’) = \kappa^{2},\] say, we deduce \(\sum 1/(y’ – z’) = 0\), or \(\sum x’^{2} – \sum y’z’ = 0\). Now apply the result of the last example.]
3. Similar triangles \(BCX\), \(CAY\), \(ABZ\) are described on the sides of a triangle \(ABC\). Show that the centres of gravity of \(ABC\), \(XYZ\) are coincident.
[We have \((x – c)/(b – c) = (y – a)/(c – a) = (z – b)/(a – b) = \lambda\), say. Express \(\frac{1}{3}(x + y + z)\) in terms of \(a\), \(b\), \(c\).]
4. If \(X\), \(Y\), \(Z\) are points on the sides of the triangle \(ABC\), such that \[BX/XC = CY/YA = AZ/ZB = r,\] and if \(ABC\), \(XYZ\) are similar, then either \(r = 1\) or both triangles are equilateral.
5. If \(A\), \(B\), \(C\), \(D\) are four points in a plane, then \[AD \cdot BC \leq BD \cdot CA + CD \cdot AB.\]
[Let \(z_{1}\), \(z_{2}\), \(z_{3}\), \(z_{4}\) be the complex numbers corresponding to \(A\), \(B\), \(C\), \(D\). Then we have identically \[(x_{1} – x_{4})(x_{2} – x_{3}) + (x_{2} – x_{4})(x_{3} – x_{1}) + (x_{3} – x_{4})(x_{1} – x_{2}) = 0.\] Hence \[\begin{aligned} |(x_{1} – x_{4})(x_{2} – x_{3})| &= |(x_{2} – x_{4})(x_{3} – x_{1}) + (x_{3} – x_{4})(x_{1} – x_{2})|\\ &\leq |(x_{2} – x_{4})(x_{3} – x_{1})| + |(x_{3} – x_{4})(x_{1} – x_{2})|.]\end{aligned}\]
6. Deduce Ptolemy’s Theorem concerning cyclic quadrilaterals from the fact that the cross ratios of four concyclic points are real. [Use the same identity as in the last example.]
7. If \(z^{2} + z’^{2} = 1\), then the points \(z\), \(z’\) are ends of conjugate diameters of an ellipse whose foci are the points \(1\), \(-1\). [If \(CP\), \(CD\) are conjugate semi-diameters of an ellipse and \(S\), \(H\) its foci, then \(CD\) is parallel to the external bisector of the angle \(SPH\), and \(SP \cdot HP = CD^{2}\).]
8. Prove that \(|a + b|^{2} + |a – b|^{2} = 2\{|a|^{2} + |b|^{2}\}\). [This is the analytical equivalent of the geometrical theorem that, if \(M\) is the middle point of \(PQ\), then \(OP^{2} + OQ^{2} = 2OM^{2} + 2MP^{2}\).]
9. Deduce from Ex. 8 that \[|a + \sqrt{a^{2} – b^{2}}| + |a – \sqrt{a^{2} – b^{2}}| = |a + b| + |a – b|.\]
[If \(a + \sqrt{a^{2} – b^{2}} = z_{1}\), \(a – \sqrt{a^{2} – b^{2}} = z_{2}\), we have \[|z_{1}|^{2} + |z_{2}|^{2} = \tfrac{1}{2}|z_{1} + z_{2}|^{2} + \tfrac{1}{2}|z_{1} – z_{2}|^{2} = 2|a|^{2} + 2|a^{2} – b^{2}|,\] and so \[(|z_{1}| + |z_{2}|)^{2} = 2\{|a|^{2} + |a^{2} – b^{2}| + |b|^{2}\} = |a + b|^{2} + |a – b|^{2} + 2|a^{2} – b^{2}|.\]Another way of stating the result is: if \(z_{1}\) and \(z_{2}\) are the roots of \(\alpha z^{2} + 2\beta z + \gamma = 0\), then \[|z_{1}| + |z_{2}| = (1/|\alpha|) \{(|-\beta +{\sqrt{\alpha\gamma}}|) + (|-\beta – {\sqrt{\alpha\gamma}}|)\}.]\]
10. Show that the necessary and sufficient conditions that both the roots of the equation \(z^{2} + az + b = 0\) should be of unit modulus are \[|a| \leq 2,\quad |b| = 1,\quad \operatorname{am} b = 2\operatorname{am} a.\]
[The amplitudes have not necessarily their principal values.]
11. If \(x^{4} + 4a_{1}x^{3} + 6a_{2}x^{2} + 4a_{3}x + a_{4} = 0\) is an equation with real coefficients and has two real and two complex roots, concyclic in the Argand diagram, then \[a_{3}^{2} + a_{1}^{2}a_{4} + a_{2}^{3} – a_{2}a_{4} – 2a_{1}a_{2}a_{3} = 0.\]
12. The four roots of \(a_{0}x^{4} + 4a_{1}x^{3} + 6a_{2}x^{2} + 4a_{3}x + a_{4} = 0\) will be harmonically related if \[a_{0}a_{3}^{2} + a_{1}^{2}a_{4} + a_{2}^{3} – a_{0}a_{2}a_{4} – 2a_{1}a_{2}a_{3} = 0.\]
[Express \(Z_{23, 14} Z_{31, 24} Z_{12, 34}\), where \(Z_{23, 14} = (z_{1} – z_{2}) (z_{3} – z_{4}) + (z_{1} -z_{3}) (z_{2} – z_{4})\) and \(z_{1}\), \(z_{2}\), \(z_{3}\), \(z_{4}\) are the roots of the equation, in terms of the coefficients.]
13. Imaginary points and straight lines. Let \(ax + by + c = 0\) be an equation with complex coefficients (which of course may be real in special cases).
If we give \(x\) any particular real or complex value, we can find the corresponding value of \(y\). The aggregate of pairs of real or complex values of \(x\) and \(y\) which satisfy the equation is called an imaginary straight line; the pairs of values are called imaginary points, and are said to lie on the line. The values of \(x\) and \(y\) are called the coordinates of the point \((x, y)\). When \(x\) and \(y\) are real, the point is called a real point: when \(a\), \(b\), \(c\) are all real (or can be made all real by division by a common factor), the line is called a real line. The points \(x = \alpha + \beta i\), \(y = \gamma + \delta i\) and \(x = \alpha – \beta i\), \(y = \gamma – \delta i\) are said to be conjugate; and so are the lines \[(A + A’i)x + (B + B’i)y + C + C’i = 0,\quad (A – A’i)x + (B – B’i)y + C – C’i = 0.\]
Verify the following assertions:—every real line contains infinitely many pairs of conjugate imaginary points; an imaginary line in general contains one and only one real point; an imaginary line cannot contain a pair of conjugate imaginary points:—and find the conditions (a) that the line joining two given imaginary points should be real, and (b) that the point of intersection of two imaginary lines should be real.
14. Prove the identities \[\begin{gathered} (x + y + z) (x + y\omega_{3} + z\omega_{3}^{2}) (x + y\omega_{3}^{2} + z\omega_{3}) = x^{3} + y^{3} + z^{3} – 3xyz,\\ (x + y + z) (x + y\omega_{5} + z\omega_{5}^{4}) (x + y\omega_{5}^{2} + z\omega_{5}^{3}) (x + y\omega_{5}^{3} + z\omega_{5}^{2}) (x + y\omega_{5}^{4} + z\omega_{5})\\ = x^{5} + y^{5} + z^{5} – 5x^{3}yz + 5xy^{2}z^{2}.\end{gathered}\]
15. Solve the equations \[x^{3} – 3ax + (a^{3} + 1) = 0,\quad x^{5} – 5ax^{3} + 5a^{2}x + (a^{5} + 1) = 0.\]
16. If \(f(x) = a_{0} + a_{1}x + \dots + a_{k}x^{k}\), then \[\{f(x) + f(\omega x) + \dots + f(\omega^{n-1}x)\}/n = a_{0} + a_{n}x^{n} + a_{2n}x^{2n} + \dots + a_{\lambda n}x^{\lambda n},\] \(\omega\) being any root of \(x^{n} = 1\) (except \(x = 1\)), and \(\lambda n\) the greatest multiple of \(n\) contained in \(k\). Find a similar formula for \(a_{\mu} + a_{\mu+n}x^{n} + a_{\mu+2n}x^{2n} + \dots\).
17. If \[(1 + x)^{n} = p_{0} + p_{1}x + p_{2}x^{2} + \dots,\] \(n\) being a positive integer, then \[p_{0} – p_{2} + p_{4} – \dots = 2^{\frac{1}{2} n} \cos\tfrac{1}{4}n\pi,\quad p_{1} – p_{3} + p_{5} – \dots = 2^{\frac{1}{2} n} \sin\tfrac{1}{4}n\pi.\]
18. Sum the series \[\frac{x}{2! {(n – 2)!}} + \frac{x^{2}}{5! {(n – 5)!}} + \frac{x^{3}}{8! {(n – 8)!}} + \dots + \frac{x^{n/3}}{{(n – 1)!}},\] \(n\) being a multiple of \(3\).
19. If \(t\) is a complex number such that \(|t| = 1\), then the point \(x = (at + b)/(t – c)\) describes a circle as \(t\) varies, unless \(|c| = 1\), when it describes a straight line.
20. If \(t\) varies as in the last example then the point \(x = \frac{1}{2}\{at + (b/t)\}\) in general describes an ellipse whose foci are given by \(x^{2} = ab\), and whose axes are \(|a| + |b|\) and \(|a| – |b|\). But if \(|a| = |b|\) then \(x\) describes the finite straight line joining the points \(-\sqrt{ab}\), \(\sqrt{ab}\).
21. Prove that if \(t\) is real and \(z = t^{2} – 1 + \sqrt{t^{4} – t^{2}}\), then, when \(t^{2} < 1\), \(z\) is represented by a point which lies on the circle \(x^{2} + y^{2} + x = 0\). Assuming that, when \(t^{2} > 1\), \(\sqrt{t^{4} – t^{2}}\) denotes the positive square root of \(t^{4} – t^{2}\), discuss the motion of the point which represents \(z\), as \(t\) diminishes from a large positive value to a large negative value.
22. The coefficients of the transformation \(z = (aZ + b)/(cZ + d)\) are subject to the condition \(ad – bc = 1\). Show that, if \(c \neq 0\), there are two fixed points \(\alpha\), \(\beta\), points unaltered by the transformation, except when \((a + d)^{2} = 4\), when there is only one fixed point \(\alpha\); and that in these two cases the transformation may be expressed in the forms \[\frac{z – \alpha}{z – \beta} = K\frac{Z – \alpha}{Z – \beta},\quad \frac{1}{z – \alpha} = \frac{1}{Z – \alpha} + K.\]
Show further that, if \(c = 0\), there will be one fixed point \(\alpha\) unless \(a = d\), and that in these two cases the transformation may be expressed in the forms \[z – \alpha = K(Z – \alpha),\quad z = Z + K.\]
Finally, if \(a\), \(b\), \(c\), \(d\) are further restricted to positive integral values (including zero), show that the only transformations with less than two fixed points are of the forms \((1/z) = (1/Z) + K\), \(z = Z + K\).
23. Prove that the relation \(z = (1 + Zi)/(Z + i)\) transforms the part of the axis of \(x\) between the points \(z = 1\) and \(z = -1\) into a semicircle passing through the points \(Z = 1\) and \(Z = -1\). Find all the figures that can be obtained from the originally selected part of the axis of \(x\) by successive applications of the transformation.
24. If \(z = 2Z + Z^{2}\) then the circle \(|Z| = 1\) corresponds to a cardioid in the plane of \(z\).
25. Discuss the transformation \(z = \frac{1}{2}\{Z + (1/Z)\}\), showing in particular that to the circles \(X^{2} + Y^{2} = \alpha^{2}\) correspond the confocal ellipses \[\frac{x^{2}}{\left\{\dfrac{1}{2}\left(\alpha + \dfrac{1}{\alpha}\right)\right\}^{2}} + \frac{y^{2}}{\left\{\dfrac{1}{2}\left(\alpha – \dfrac{1}{\alpha}\right)\right\}^{2}} = 1.\]
26. If \((z + 1)^{2} = 4/Z\) then the unit circle in the \(z\)-plane corresponds to the parabola \(R\cos^{2} \frac{1}{2}\Theta = 1\) in the \(Z\)-plane, and the inside of the circle to the outside of the parabola.
27. Show that, by means of the transformation \(z = \{(Z – ci)/(Z + ci)\}^{2}\), the upper half of the \(z\)-plane may be made to correspond to the interior of a certain semicircle in the \(Z\)-plane.
28. If \(z = Z^{2} – 1\), then as \(z\) describes the circle \(|z| = \kappa\), the two corresponding positions of \(Z\) each describe the Cassinian oval \(\rho_{1}\rho_{2} = \kappa\), where \(\rho_{1}\), \(\rho_{2}\) are the distances of \(Z\) from the points \(-1\), \(1\). Trace the ovals for different values of \(\kappa\).
29. Consider the relation \(az^{2} + 2hzZ + bZ^{2} + 2gz + 2fZ + c = 0\). Show that there are two values of \(Z\) for which the corresponding values of \(z\) are equal, and vice versa. We call these the branch points in the \(Z\) and \(z\)-planes respectively. Show that, if \(z\) describes an ellipse whose foci are the branch points, then so does \(Z\).
[We can, without loss of generality, take the given relation in the form \[z^{2} + 2zZ\cos\omega + Z^{2} = 1:\] the reader should satisfy himself that this is the case. The branch points in either plane are \(\csc\omega\) and \(-\csc\omega\). An ellipse of the form specified is given by \[|z + \csc\omega| + |z – \csc\omega| = C,\] where \(C\) is a constant. This is equivalent (Ex. 9) to \[|z + \sqrt{z^{2} – \csc^{2}\omega}| + |z – \sqrt{z^{2} – \csc^{2}\omega}| = C.\] Express this in terms of \(Z\).]
30. If \(z = aZ^{m} + bZ^{n}\), where \(m\), \(n\) are positive integers and \(a\), \(b\) real, then as \(Z\) describes the unit circle, \(z\) describes a hypo- or epi-cycloid.
31. Show that the transformation \[z = \frac{(a + di)Z_{0} + b}{cZ_{0} – (a – di)},\] where \(a\), \(b\), \(c\), \(d\) are real and \(a^{2} + d^{2} + bc > 0\), and \(Z_{0}\) denotes the conjugate of \(Z\), is equivalent to an inversion with respect to the circle \[c(x^{2} + y^{2}) – 2ax – 2dy – b = 0.\] What is the geometrical interpretation of the transformation when \[a^{2} + d^{2} + bc < 0?\]
32 The transformation \[\frac{1 – z}{1 + z} = \left(\frac{1 – Z}{1 + Z}\right)^{c},\] where \(c\) is rational and \(0 < c < 1\), transforms the circle \(|z| = 1\) into the boundary of a circular lune of angle \(\pi/c\).
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