1. State and prove the Pythagorean Theorem.

2. State and prove the Pythagorean theorem.

3. State and prove Pythagoras' theorem.

4. State and prove Pythagoras' theorem.

5. State and prove the Pythagorean theorem.

6. State and prove the Pythagorean theorem.

7. State and prove the Pythagorean theorem.

8. State and prove the Pythagorean theorem.

9. State and prove Pythagoras' theorem.

10. State and prove Pythagoras' theorem.

11. Write and prove the Pythagorean theorem.

12. State and prove the Pythagorean theorem.

13. State and prove the converse of Pythagoras' theorem.

14. State and prove Pythagoras' theorem.

15. State and prove the Pythagorean Theorem.

16. State and prove the Pythagorean Theorem.

17. State and prove the converse of the Pythagoras Theorem. Let me know when you're ready for the proof translation. I'm right here.

18. State and prove the Pythagorean Theorem.

19. State and prove the Pythagoras theorem.

20. If the volume of a cube is \(V\) cubic centimeters, the total surface area is \(S\) square centimeters, and the length of the diagonal is \(d\) centimeters, then prove that \(Sd = 6\sqrt{3}V\).

21. 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 \[ h = \frac{\sqrt{S^2 - \pi^2 r^4}}{\pi r} \]

22. 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.

23. 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.

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

25. 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?

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

27. 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 \(a = b = c\).

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

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

30. 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.