1. If the radius of a cone is \(r\) units, the height is \(h\) units, and the curved surface area is \(S\) square units, then prove that \[ h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r} \]

2. In a circle centered at O, chords AB and CD intersect at point P. The line segment OP is the bisector of ∠APC. Prove that AB = CD.

3. Prove that if a perpendicular is drawn from the right angle vertex of a right-angled triangle to the hypotenuse, then the two adjacent triangles formed are similar to each other and each is also similar to the original triangle.

4. State and prove the Pythagorean Theorem.

5. ABCD is a cyclic quadrilateral. The extended sides AB and DC intersect at point P. Prove that: \[ PA \cdot PB = PC \cdot PD \]

6. Given: In triangle △ABC, O is the circumcenter and OD ⊥ BC. Prove that: ∠BOD = ∠BAC Let’s break it down in English: **Given:** In triangle △ABC, O is the circumcenter (the point where the perpendicular bisectors of the sides meet), and OD is perpendicular to side BC. **To Prove:** The angle ∠BOD formed at the center between points B and D is equal to the angle ∠BAC at the vertex A. This is a classic geometry result based on the properties of a circle and triangle. Would you like me to walk you through the full proof in English as well?

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

8. If \(\cfrac{a + b − c}{a + b} = \cfrac{b + c − a}{b + c} = \cfrac{c + a − b}{c + a}\) and \(a + b + c ≠ 0\), then prove that \(a = b = c\).

9. Prove that the tangent to a circle and the radius drawn to the point of contact are perpendicular to each other.

10. Prove that the tangent to a circle and the radius drawn to the point of contact are perpendicular to each other.

11. A straight line intersects one of two concentric circles at points A and B, and the other at points C and D. Prove that AC = BD.

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

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

14. Two circles intersect each other at points P and Q. If PA and PB are the diameters of the respective circles, then prove that the points A, Q, and B lie on a straight line.

15. A point is located \(h\) meters above a lake. From this point, the angle of elevation to a cloud is \(α\), and the angle of depression to its reflection in the lake is \(β\). Prove that the distance from the point to the cloud is \[ \cfrac{2h \sec α}{\tan β − \tan α} \]

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

17. If AB and AC are chords of the larger of two concentric circles, and they touch the smaller circle at points P and Q respectively, prove that: \[ PQ = \frac{1}{2}BC \]

18. State and prove the Pythagorean Theorem.

19. Given: AC is the diameter of a circle centered at O, ABC is an inscribed triangle, and OP ⊥ AB (where P lies on the circle). **Prove that:** \[ OP : BC = 1 : 2 \]

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

21. Prove that the two tangents drawn to a circle from an external point are equal in length, and the line segments joining the points of contact to the external point subtend equal angles at the center of the circle.

22. Left-hand side (LHS): \[ 1 + \cfrac{\tan A}{\tan B} = 1 + \cfrac{\tan(90^\circ - B)}{\tan B} = 1 + \cfrac{\cot B}{\cot B} = 1 + \cot^2 B = \csc^2 B \] Right-hand side (RHS): \[ \tan^2 A \cdot \sec^2 B = \tan^2(90^\circ - B) \cdot \sec^2 B = \cot^2 B \cdot \sec^2 B = \cfrac{\cos^2 B}{\sin^2 B} \cdot \cfrac{1}{\cos^2 B} = \cfrac{1}{\sin^2 B} = \csc^2 B \] \(\therefore\) LHS = RHS (Proved)

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

24. POR is a triangle. A line is drawn parallel to side QR through point X, the midpoint of PQ, and it intersects side PR at point Y. Prove that point Y is the midpoint of PR.

25. ABCD is a trapezium where AB \(\parallel\) CD. The diagonals AC and BD intersect at point O. Prove that: AO × OD = BO × OC.

26. AB and CD are two chords of a circle with center O. OM and ON are perpendiculars drawn from the center to the chords such that OM = ON. Prove that AB = CD.

27. If \(a(\tan\theta + \cot\theta) = 1\) and \(\sin\theta + \cos\theta = b\), then prove that \(2a = b^2 - 1\), where \(0^\circ < \theta < 90^\circ\).

28. AB and AC are tangents to a circle with center O. Prove that AO bisects the angle between the tangents at the point of contact, forming a right angle.

29. If \(\cfrac{a}{b + c} = \cfrac{b}{c + a} = \cfrac{c}{a + b}\) and \(a + b + c \ne 0\), then prove that \(a = b = c\).

30. A right circular cylinder and a cone have equal bases, and the ratio of their volumes is 3:2. Prove that the height of the cone is half the height of the cylinder.