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Let's tackle these three problems one by one.

### Bài 1.30:
Cho tam giác \( ABC \) thỏa mãn \( \sin A = 2 \sin B \cos C \). Chứng minh rằng tam giác \( ABC \) cân.

**Solution:**

We are given the equation:
\[
\sin A = 2 \sin B \cos C
\]

1. Using the product-to-sum identity for trigonometric functions, we can rewrite \( 2 \sin B \cos C \):
   \[
   2 \sin B \cos C = \sin(B + C) + \sin(B - C)
   \]

2. Substituting this into the given equation, we have:
   \[
   \sin A = \sin(B + C) + \sin(B - C)
   \]

3. From the angle sum property of triangles, \( A + B + C = 180^\circ \), or \( B + C = 180^\circ - A \). Therefore, \( \sin(B + C) = \sin A \). This gives us:
   \[
   \sin A = \sin A + \sin(B - C)
   \]

4. Simplifying this, we get:
   \[
   0 = \sin(B - C)
   \]

5. Therefore, \( B = C \), which shows that \( \triangle ABC \) is isosceles (cân).

### Bài 1.31:
Cho tam giác \( ABC \) thỏa mãn \( a^4 = b^4 + c^4 \). Chứng minh:

#### (a) Tam giác \( ABC \) nhọn:
To prove the triangle is acute, we need to use the cosine rule:
\[
a^2 = b^2 + c^2 - 2bc \cos A
\]
However, the given condition is \(

Làm 3 bài này: Bài 1.30: Cho tam giác ABC thỏa sin A = 2sin B cos C. Chứng minh tam giác ABC cân. Bài 1.31: Cho tam giác ABC thỏa a^4 = b^4 + c^4. Chứng minh (a) Tam giác ABC nhọn (b) 2 sin^2 A = tan A · tan B. Bài 1.32: Cho tam giác ABC, gọi ra, rb, rc là bán kính đường tròn bàng tiếp góc A, B, C. Chứng minh các hệ thức liên quan đến bán kính và tổng các bán kính.

Solve problems involving triangle properties and trigonometric identities, including proving the triangle is isosceles, acute, and formulas for the radii of excircles. Suitable for high school students studying advanced geometry and trigonometry.

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