1. From vertex \(A\) of \(\triangle ABC\), a perpendicular \(AD\) is drawn to side \(BC\). If \(\frac{BD}{DA} = \frac{DA}{DC}\), then prove that \(\triangle ABC\) is a right-angled triangle.

2. 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?

3. In right-angled triangle \( \triangle ABC \), where \( \angle A = 90^\circ \), a perpendicular \( AD \) is drawn from point \( A \) to the hypotenuse \( BC \). Prove that: \[ \frac{\text{Area of } \triangle ABC}{\text{Area of } \triangle ACD} = \frac{BC^2}{AC^2} \]

4. From the vertex A of triangle \( \triangle ABC \), a perpendicular AD is drawn to the side BC. If \[ \frac{BD}{DA} = \frac{DA}{DC} \] then prove that triangle ABC is a right-angled triangle.

5. In right-angled triangle ABC, \(\angle A\) is the right angle. A perpendicular AD is drawn from point A to the hypotenuse BC. Prove that: \[ \frac{∆ABC}{∆ACD} = \frac{BC^2}{AC^2} \]

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. 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.

8. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar, and each of these triangles is similar to the original triangle.

9. In \( \triangle ABC \), the vertex \( A \) has a perpendicular \( AD \) drawn to the side \( BC \). If \[ \frac{BD}{DA} = \frac{DA}{DC} \] then prove that \( \triangle ABC \) is a right-angled triangle.

10. Prove that if a perpendicular is drawn from the right-angled vertex of any right-angled triangle to the hypotenuse, then the two resulting triangles on either side of the perpendicular are similar to each other and each is also similar to the original triangle.

11. Prove that if a perpendicular is drawn from the right-angled vertex of a right-angled triangle to the hypotenuse, then the two triangles formed on either side of this perpendicular are similar to each other and each is similar to the original triangle.

12. In a right-angled triangle, if the ratio of the perpendicular (opposite side) to the hypotenuse with respect to a positive acute angle \(\theta\) is \(12 : 13\), then determine the ratio of the perpendicular to the base and the ratio of the hypotenuse to the base, and verify that \( \sec^2\theta = 1 + \tan^2\theta \).