1. If \(a\propto b\) and \(b\propto c\), prove that: \[ a^2+b^2+c^2 \propto ab+bc+ca \]

2. If \(bcx = cay = abz\), then prove that \[ \frac{ax + by}{a^2 + b^2} = \frac{by + cz}{b^2 + c^2} = \frac{cz + ax}{c^2 + a^2} \]

3. If \(a + b + c = 0\), then prove that \[ \frac{bc}{2a^2 + bc} + \frac{ca}{2b^2 + ca} + \frac{ab}{2c^2 + ab} = 1 \]

4. If \(a + b + c = 0\), then prove that: \[ \cfrac{1}{2a^2 + bc} + \cfrac{1}{2b^2 + ca} + \cfrac{1}{2c^2 + ab} = 0 \]

5. If \(a, b, c, d\) are in continued proportion, then prove that \[ (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2 \]

6. If \(a, b, c, d\) are in continued geometric progression, prove that \[(a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = (ab + bc + cd)^2\]

7. If \(\frac{a^2}{b + c} = \frac{b^2}{c + a} = \frac{c^2}{a + b} = 1\), then prove that \[\frac{1}{1 + a} + \frac{1}{1 + b} + \frac{1}{1 + c} = 1\]

8. ABCD is a rectangle and O is any point inside the rectangle. Prove that: \[ OA^2 + OC^2 = OB^2 + OD^2 \]

9. If \[ \frac{by + cz}{b^2 + c^2} = \frac{cz + ax}{c^2 + a^2} = \frac{ax + by}{a^2 + b^2} \] then prove that \[ \frac{x}{a} = \frac{y}{b} = \frac{z}{c} \]

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

11. If \(a^2 = by + cz\), \(b^2 = cz + ax\), and \(c^2 = ax + by\), then prove that \[ \frac{x}{a + x} + \frac{y}{b + y} + \frac{z}{c + z} = 1 \]

12. If \(a, b, c\) are in continued geometric progression (i.e., in successive proportion), then prove that \[ \cfrac{1}{b^2} = \cfrac{1}{b^2 - a^2} + \cfrac{1}{b^2 - c^2} \]

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

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

15. If \(a : b = b : c\), then prove that \[ a^2 b^2 c^2 \left( \frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]

16. If \(\cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} + \cfrac{c^2}{a+b} = 1\), then prove that \[ \cfrac{1}{1+a} + \cfrac{1}{1+b} + \cfrac{1}{1+c} = 1 \]

17. If \(a:b = b:c\), then prove that \[ a^2b^2c^2\left(\cfrac{1}{a^3}+\cfrac{1}{b^3}+\cfrac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

18. If \[ \cfrac{a^2}{b+c} = \cfrac{b^2}{c+a} = \cfrac{c^2}{a+b} = 1 \] then prove that \[ \cfrac{1}{a+1} + \cfrac{1}{b+1} + \cfrac{1}{c+1} = 1. \]

19. If \(a : b = b : c\), then prove that \[ \left(\cfrac{a + b}{b + c}\right)^2 = \cfrac{a^2 + b^2}{b^2 + c^2} \]

20. If \(a : b = b : c\), then prove that \[ a^2b^2c^2\left(\cfrac{1}{a^3} + \cfrac{1}{b^3} + \cfrac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

21. If \(\cfrac{m}{a} = \cfrac{n}{b}\), then prove that \[ (m^2 + n^2)(a^2 + b^2) = (am + bn)^2 \]

22. If \(\cfrac{a}{b} = \cfrac{x}{y}\), then prove that \[ (a + b)(a^2 + b^2)x^3 = (x + y)(x^2 + y^2)a^3 \]

23. If \[ \cfrac{x + y}{3a - b} = \cfrac{y + z}{3b - c} = \cfrac{z + x}{3c - a} \] then prove that \[ \cfrac{x + y + z}{a + b + c} = \cfrac{ax + by + cz}{a^2 + b^2 + c^2} \]

24. If \( \tan 9^\circ = \frac{a}{b} \), then prove that \[ \frac{\sec^2 81^\circ}{1 + \cot^2 81^\circ} = \frac{b^2}{a^2} \]

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

26. If \(a, b, c\) are in continued geometric progression, then prove that \[ a^2 b^2 c^2 \left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3} \right) = a^3 + b^3 + c^3 \]

27. If ABCD is a cyclic quadrilateral inscribed in a circle with center O, then prove that: \[ AB + CD = BC + DA \]

28. In triangle △ABC, ∠C = 90°, and if BC = \(m\) and AC = \(n\), then prove that: \[ m \sin A + n \sin B = \sqrt{m^2 + n^2} \]

29. If \(a, b, c\) are in continued geometric progression, prove that \[ a^2b^2c^2\left(\frac{1}{a^3} + \frac{1}{b^3} + \frac{1}{c^3}\right) = a^3 + b^3 + c^3 \]

30. If \( a = \frac{\sqrt{5} + 1}{\sqrt{5} - 1} \) and \( b = \frac{\sqrt{5} - 1}{\sqrt{5} + 1} \), then find the value of \[ \frac{a^2 + ab + b^2}{a^2 - ab + b^2} \]