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

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

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

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

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

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

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

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

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

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

11. In \(\triangle ABC\), points \(D\) and \(E\) lie on sides \(AB\) and \(AC\) respectively, such that \(DE \parallel BC\) and \(AD : DB = 3 : 1\). If \(EA = 3.3\) cm, determine the length of \(AC\).

(a) 1.1 cm (b) 4 cm (c) 4.4 cm (d) 5.5 cm

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

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

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

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

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

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

20. If \(x : a = y : b = z : c\), then prove that \(\cfrac{x^3}{a^2} + \cfrac{y^3}{b^2} + \cfrac{z^3}{c^2} = \cfrac{(x + y + z)^3}{(a + b + c)^2}\)

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

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

23. If \(a : b = c : d = e : f\), then prove that each of these ratios is equal to \(\cfrac{5a - 7c - 13e}{5b - 7d - 13f}\)

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

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

26. In the right-angled triangle \(ABC\), where \(\angle B = 90^\circ\), point \(D\) lies on \(AB\) such that the ratio \(AB : BC : BD = \sqrt{3} : 1 : 1\). Find the measure of \(\angle ACD\).

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

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

29. In triangle \(∆ABC\), the medians \(BE\) and \(CF\) intersect each other at point \(G\), and the line segment \(FE\) intersects the line segment \(AG\) at point \(O\). Prove that: \(AO = 3 \times OG\).

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