1. If in triangle ABC, AD is perpendicular to side BC and \(AD^2 = BD \cdot DC\), prove that \(\angle BAC\) is a right angle.

2. In triangle ABC, let AD be perpendicular to BC and \(AD^2 = BD \cdot DC\). Prove that \(\angle BAC\) is a right angle.

3. From vertex A of triangle \(\triangle ABC\), a perpendicular AD is drawn to the base BC, intersecting BC at point D. If \(AD^2 = BD \cdot CD\), prove that triangle ABC is a right-angled triangle and that \(\angle A = 90^\circ\).

4. 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?

5. 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.

6. 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.

7. 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.

8. 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.

9. 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.

10. The circumcenter of triangle ΔABC is O, and OD is perpendicular to side BC. Prove that \(\angle\)BOD = \(\angle\)BAC.

11. 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.

12. 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: \[ \frac{∆ABC}{∆ACD} = \frac{BC^2}{AC^2} \]