1. If \(x \propto y\) and \(y \propto z\), then show that \[ x^2 + y^2 + z^2 \propto xy + yz + zx \]

2. If \(x \propto y\) and \(y \propto z\), then prove that \[ x^2 + y^2 + z^2 \propto xy + yz + zx \]

3. If \(x \propto y\) and \(y \propto z\), then show that \(x + y \propto z\).

4. If \(x \propto y\) and \(y \propto z\), then prove that \(x^2 + y^2 + z^2 \propto xy + yz + zx\)

5. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that: \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

6. If \(\left(\frac{1}{x} - \frac{1}{y}\right) \propto \frac{1}{x - y}\), then show that \((x^2 + y^2) \propto xy\).

7. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} \]

8. If \(x \propto y\) and \(y \propto z\), then \((x + y + z) \propto (\sqrt{yz} + \sqrt{zx} + \sqrt{xy})\).

9. If \(a \propto b\) and \(b \propto c\), then show that \[ a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3) \]

10. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then prove that \[ \frac{x^2}{1 - a^2} = \frac{y^2}{1 - b^2} = \frac{z^2}{1 - c^2} \]

11. If \(x \propto y\) and \(y \propto z\), prove that \(x^3 + y^3 + z^3 \propto xyz\).

12. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2 =\) constant.

13. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2\) is constant.

14. If \(\cfrac{x}{y} \propto x + y\) and \(\cfrac{y}{x} \propto x - y\), then show that \(x^2 - y^2\) is a constant.

15. If \(a \propto b\) and \(b \propto c\), then show that \[a^3b^3 + b^3c^3 + c^3a^3 \propto abc(a^3 + b^3 + c^3)\]

16. When \(x + y \propto z\) with \(y\) constant, and \(z + x \propto y\) with \(z\) constant, show that \(x + y + z \propto yz\) when both \(y\) and \(z\) vary.

17. If \(x \propto y\) and \(x \propto z\), then \((y + z) \propto\) ---- If \(x\) is proportional to \(y\) and \(x\) is proportional to \(z\), then \((y + z)\) is proportional to -

18. If \(x \alpha y\) and \(y \alpha z\), then prove that: \[ x^2 + y^2 + z^2 \alpha xy + yz + zx \]

19. If \((u^2 + v^2) \propto (x^2 + y^2)\) and\(uv \propto xy\), then show that\((u + v) \propto (x + y)\) when \(\frac{u}{v} + \frac{v}{u} = \frac{x}{y} + \frac{y}{x}\)."

20. If \(2x^2 + 3y^2 \propto xy\), then show that \(x \propto y\).

21. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(y = \frac{\sqrt{3} - 1}{\sqrt{3} + 1}\), then show that \[ \frac{x^2 + y^2}{x^2 - y^2} = \frac{7\sqrt{3}}{12} \]

22. If \((b + c - a)x = (c + a - b)y = (a + b - c)z = 2\), then show that \[ \left(\frac{1}{x} + \frac{1}{y}\right)\left(\frac{1}{y} + \frac{1}{z}\right)\left(\frac{1}{z} + \frac{1}{x}\right) = abc \]

23. If \(\left(x^3 - \frac{1}{y^3}\right) \propto \left(x^3 + \frac{1}{y^3}\right)\), then show that \(x \propto \frac{1}{y}\).

24. If \(x + y \propto x - y\), then show that \(x^3 + y^3 \propto x^3 - y^3\).

25. If \(x \propto y\) and \(x \propto z\), then \((y + z) \propto\) ________.

26. If \(\frac{x}{y} \propto x + y\) and \(\frac{y}{x} \propto x - y\), then show that \((x^2 - y^2)\) is a constant quantity.

27. If \(\frac{1}{x^3} + \frac{1}{y^3} \propto \frac{1}{x^3 + y^3}\), then show that \(x \propto y\).

28. If \(x \propto z\) and \(y \propto z\), then \(x^2 + y^2 \propto z^2\).

29. If \(\cfrac{1}{x}+\cfrac{1}{y}\propto \cfrac{1}{x+y}\), then show that \(x\propto y\).

30. If \(x = cy + bz\), \(y = az + cx\), and \(z = bx + ay\), then show that \(x^2 : y^2 = (1 - a^2) : (1 - b^2)\).