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

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

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

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

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

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

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

10. If \(x \propto y\) and \(y \propto z\), then show that \[ \frac{x}{yz} + \frac{y}{zx} + \frac{z}{xy} \propto \frac{1}{x} + \frac{1}{y} + \frac{1}{z} \]

11. If \(x + y ∝ x - y\), then show that \(x^2 + y^2 ∝ xy\).

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

13. If \(y\) is the geometric mean between \(x\) and \(z\), then what is the geometric mean between \(x^2 + y^2\) and \(y^2 + z^2\)?

14. If \((x + y) ∝ (x - y)\), then \((x^2 + y^2) ∝ xy\).

15. If \(a ∝ b\) and \(b ∝ c\), then prove that \(a^3 + b^3 + c^3 ∝ 3abc\).

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

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

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

19. If \(x ∝ y\), then show that \(x + y ∝ x - y\).

20. If \(x = a \cos\theta + b \sin\theta\) and \(y = a \sin\theta - b \cos\theta\), then prove that \(x^2 + y^2 = a^2 + b^2\).