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

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

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

4. If \(x : a = y : b = z : c\), then show that \(\frac{x^3}{a^3} + \frac{y^3}{b^3} + \frac{z^3}{c^3} = \frac{3xyz}{abc}\).

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

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

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

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

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

10. If the roots of the quadratic equation \(ax^2 + bx + c = 0\) are in the ratio \(1 : s\), then show that \[ \frac{(s + 1)^2}{s} = \frac{b^2}{ac} \]

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

12. If \(a:b = b:c\), show that \((a+b+c)(a-b+c) = a^2 + b^2 + c^2\).

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

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

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

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

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

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

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

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

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 \(A = B = 45^\circ\), then let us verify that: \(\sin (A + B) = \sin A \cos B + \cos A \sin B\)

23. If \(A = B = 45^\circ\), then let us verify that: \(\cos (A + B) = \cos A \cos B - \sin A \sin B\)

24. If \(x : a = y : b = z : c\), then show that \(\cfrac{x^3}{a^3} + \cfrac{y^3}{b^3} + \cfrac{z^3}{c^3} = \cfrac{3xyz}{abc}\).

25. If \(x : a = y : b = z : c\), then show that \[ \frac{x^3 + y^3 + z^3}{a^3 + b^3 + c^3} = \frac{xyz}{abc} \]

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

27. 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: \(a + c = 2b\).

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

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

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