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The image presents a geometry problem with the following description:

**Problem:**
Let \( D \) be a point on side \( BC \) of triangle \( ABC \) such that the incircles of triangles \( ABD \) and \( ADC \) have the same radius. If \( AD = h \), then the area of triangle \( ABC \) is:

The options provided are:

a) \( A = h^2 \cdot \tan\left(\frac{\theta}{2}\right) \)

b) \( A = h^2 \cdot \cot\left(\frac{\theta}{2}\right) \)

c) \( A = h^2 \cdot \sin\left(\frac{\theta}{2}\right) \)

d) \( A = h^2 \cdot \cos\left(\frac{\theta}{2}\right) \)

e) \( A = h^2 \)

### Explanation:
This problem relates to properties of incircles and uses trigonometric relationships, where we must calculate the area of the triangle \( ABC \) based on the given conditions.

The fact that the incircles of \( ABD \) and \( ADC \) share the same radius suggests that there is symmetry in how the sides of these triangles are positioned, which can lead to specific formulas involving the angles and sides.

To proceed with the solution, we would typically analyze the relationship between the angle at \( A \) (denoted as \( \theta \)), the side \( BC \), and how the given height \( h \) is related to the area of the entire triangle.

Would you like me to proceed by solving this problem step by step, or would you prefer more specific details on any part of the question? 

### Related Questions:
1. How do incircles relate to the area of triangles?
2. What is the relationship between triangle height and area?
3. How can we use trigonometric identities to calculate triangle areas?
4. What are the properties of triangles with equal incircle radii?
5. How can we find the area of a triangle using angle bisectors?

### Tip:
When triangles have shared properties like equal incircle radii, leveraging symmetry and geometric theorems can simplify the solution process.

Seja D um ponto sobre o lado BC do triângulo ABC, tal que os círculos inscritos nos triângulos ABD e ADC tenham o mesmo raio. Se AD = h, a área do triângulo ABC é igual a:

Learn how to calculate the area of triangle ABC when the incircles of triangles ABD and ADC have the same radius. This problem involves geometry and trigonometric relationships and is suitable for high school students.

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