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

2. Exactly — well explained. Since triangle ABC is isosceles with AC = BC, and AB² equals the sum of the squares of AC and BC, this satisfies the converse of the Pythagorean theorem. That confirms ∠C is a right angle, measuring **90 degrees**. Want to try applying this to a new triangle scenario? I’ve got a few twists if you’re up for it!

(a) 9 meters (b) 10 meters (c) 11meters (d) 12 meters

3. 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 \(Sd = 6\sqrt{3}V\).

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

5. 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.

6. 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.

7. State and prove the Pythagorean Theorem.

8. 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?

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. 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.

13. 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 \[ \cfrac{2h \sec α}{\tan β − \tan α} \]

14. 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: \[ PQ = \frac{1}{2}BC \]

15. State and prove the Pythagorean Theorem.

16. 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:** \[ OP : BC = 1 : 2 \]

17. 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.

18. 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.

19. 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.

20. 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.

21. 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.

22. If \(x = \frac{\sqrt{3} + 1}{\sqrt{3} - 1}\) and \(xy = 1\), prove that the simplified value of \(\frac{x^2}{y} + \frac{y^2}{x}\) is 52.

23. The centers of two circles are P and Q. The circles intersect at points A and B. Through point A, two straight lines parallel to the line segment PQ intersect the circles at points C and D. Prove that CD = 2PQ.

24. From an external point \(A\), two tangents are drawn to a circle centered at \(O\), touching the circle at points \(B\) and \(C\) respectively. Prove that the line \(AO\) is the perpendicular bisector of the line segment \(BC\).

25. Given: In a circle with center O, MN is any chord that is not a diameter, and AC is a diameter. To Prove: AC > MN, i.e., the diameter is the longest chord of a circle. Construction: From the center O, draw a perpendicular OD to the chord MN. Join O and M. Proof: OM > MD [∵ Triangle OMD is a right-angled triangle and OM is the hypotenuse] Or, OA > MD [∵ OA = OM, both are radii of the same circle] Or, ½AC > ½MN Or, AC > MN Therefore, the diameter is the longest chord of a circle.

26. Let \(x, y, z\) be three variables such that the value of \(y + z - x\) is constant and \((x + y - z)(z + x - y) \propto yz\). Prove that \((x + y + z) \propto yz\).

27. In a circle, chords AB and AC are equal. Prove that the bisector of ∠BAC passes through the center of the circle.

28. If the roots of the quadratic equation \((1+m^2)x^2 + 2mcx + (c^2 - a^2) = 0\) are real and equal, prove that \(c^2 = a^2(1+m^2)\).

29. If \(h, c, v\) represent the height, curved surface area, and volume of a cone, respectively, then prove that \(3\pi v h^3 - c^2 h^2 + 9v^2 = 0\).

30. If \(\alpha\) and \(\beta\) are the roots of the equation \(x^2+7x+3=0\), prove that: \[ \alpha^3+\beta^3+7(\alpha^2+\beta^2)+3(\alpha+\beta)=0 \]