1. If \(a : b = c : d\), prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).

2. If \(a:b=c:d\), then prove that \((a^2+c^2)(b^2+d^2)=(ab+cd)^2\).

3. If \(a : b = c : d\), prove that \((a^2 + c^2)(b^2 + d^2) = (ab + cd)^2\).

4. If \(a : b = c : d\), then prove that \((a^2 + ab + b^2) : (a^2 - ab + b^2) = (c^2 + cd + d^2) : (c^2 - cd + d^2)\)

5. If \(a : b = c : d\), then prove that \((a^2 + b^2) : (a^2 - b^2) = (ac + bd) : (ac - bd)\)

6. If \(a : b = c : d = e : f\), then prove that \((a^2 + c^2 + e^2)(b^2 + d^2 + f^2) = (ab + cd + ef)^2\)

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

8. If \(a : b = b : c\), then prove that \(\cfrac{abc(a+b+c)^3}{(ab+bc+ca)^3} = 1\).

9. If \(a : b = c : d\), then prove that \(\sqrt{a^2 + c^2} : \sqrt{b^2 + d^2} = (pa + qc) : (pb + qd)\)

10. If \(a : b = b : c\), then prove that \[ \cfrac{abc(a + b + c)^3}{(ab + bc + ca)^3} = 1 \]

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

12. Translate of your statement in English: If the roots of the equation \((b - c)x^2 + (c - a)x + (a - b) = 0\) are equal, then prove that: \(a + c = 2b\).

13. If \(a : b = b : c\), then prove that \[ (a + b)^2 : (b + c)^2 = a : c \]

14. If \(a : b = b : c\), then show that \[abc (a + b + c)^3 = (ab + bc + ca)^3\]

15. If \(a : b = b : c\), then prove that \((a + b + c)(a - b + c) = a^2 + b^2 + c^2\).

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

17. If a rectangular parallelepiped has volume \(v\), and its length, width, and height are \(a, b, c\) respectively, with a total surface area \(s\), prove that: \(\cfrac{1}{v}=\cfrac{2}{3}\left(\cfrac{1}{a}+\cfrac{1}{b}+\cfrac{1}{c}\right)\).

18. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:p\), prove that \(\cfrac{(p+1)^2}{p} = \cfrac{b^2}{ac}\).

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

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

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

22. If the roots of the quadratic equation \(ax^2+bx+c=0\) are in the ratio \(1:2\), then prove that \(2b^2=9ac\).

23. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then prove that \[ \frac{(r + 1)^2}{r} = \frac{b^2}{ac} \]

24. If \(x : a = y : b = z : c\), then prove that \((a^2 + b^2 + c^2)(x^2 + y^2 + z^2) = (ax + by + cz)^2\)

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

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

27. Assume, \(\cfrac{a}{b} = \cfrac{b}{c} = \cfrac{c}{d} = k\) (where \(k \ne 0\)) So, \(a = bk,\; b = ck,\; c = dk\) \(\therefore b = dk \cdot k = dk^2\) and \(a = dk^2 \cdot k = dk^3\)

Left-hand side \[ = (a^2 + b^2 + c^2)(b^2 + c^2 + d^2) = \{(dk^3)^2 + (dk^2)^2 + (dk)^2\} \cdot \{(dk^2)^2 + (dk)^2 + d^2\} \] \[ = \{d^2k^6 + d^2k^4 + d^2k^2\} \cdot \{d^2k^4 + d^2k^2 + d^2\} = d^2k^2(k^4 + k^2 + 1) \cdot d^2(k^4 + k^2 + 1) \] \[ = d^4k^2(k^4 + k^2 + 1)^2 \] Right-hand side \[ = (ab + bc + cd)^2 = (dk^3 \cdot dk^2 + dk^2 \cdot dk + dk \cdot d)^2 = (d^2k^5 + d^2k^3 + d^2k)^2 = \{d^2k(k^4 + k^2 + 1)\}^2 = d^4k^2(k^4 + k^2 + 1)^2 \] ∴ Left-hand side = Right-hand side (Hence Proved)

28. If \(A\) and \(B\) are complementary angles, prove that: \[ \frac{\sin A + \sin B}{\cos B - \cos A} = \frac{\cos A + \cos B}{\sin A - \sin B} \]

29. If \(a : b = c : d\), then prove: \[ \frac{a + b}{a - b} = \frac{\sqrt{ac + ad + bc + bd}}{\sqrt{ac - ad - bc + bd}} \]

30. If the ratio of the roots of the quadratic equation \(ax^2 + bx + c = 0\) is \(1 : r\), then prove that \(\frac{(r + 1)^2}{r} = \frac{b^2}{ac}\).