Let a,b,c be the sides of triangle whose perimeter is P and area is A, then

Options

A. `p^(3)le 27 (b+c-a) (c+a-b) (a+b-c)`
B. `p^(2)le 3 (a^(2)+b^(2) +c^(2))`
C. `a^(2)+ b^(2) +c^(2) ge 4 sqrt3A`
D. `p^(4) le 25 ltA`

Answer

To solve the problem, we need to analyze the relationships between the sides \( a, b, c \) of a triangle, its perimeter \( P \), and its area \( A \). We will derive inequalities that are true for any triangle based on these parameters.

### Step-by-Step Solution:

1. **Understanding the Perimeter and Area**:
- The perimeter \( P \) of the triangle is given by:
\[ P = a + b + c \]
- The area \( A \) can be calculated using Heron's formula:
\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]
where \( s = \frac{P}{2} = \frac{a+b+c}{2} \).

2. **Using the Triangle Inequalities**:
- For any triangle, the following inequalities must hold:
\[ a + b > c, \quad b + c > a, \quad c + a > b \]
- This implies that the differences \( b+c-a \), \( c+a-b \), and \( a+b-c \) are all positive.

3. **Applying the Arithmetic Mean-Geometric Mean (AM-GM) Inequality**:
- By applying the AM-GM inequality to the three positive terms \( b+c-a \), \( c+a-b \), and \( a+b-c \), we have:
\[ \frac{(b+c-a) + (c+a-b) + (a+b-c)}{3} \geq \sqrt[3]{(b+c-a)(c+a-b)(a+b-c)} \]
- Simplifying the left-hand side gives:
\[ \frac{P}{3} \geq \sqrt[3]{(b+c-a)(c+a-b)(a+b-c)} \]

4. **Cubing Both Sides**:
- Cubing both sides of the inequality results in:
\[ \left(\frac{P}{3}\right)^3 \geq (b+c-a)(c+a-b)(a+b-c) \]
- This can be rewritten as:
\[ \frac{P^3}{27} \geq (b+c-a)(c+a-b)(a+b-c) \]

5. **Finding the Area in Terms of the Sides**:
- The area \( A \) can also be expressed in terms of the sides using the formula:
\[ A^2 = s(s-a)(s-b)(s-c) \]
- This can be used to derive inequalities relating \( A \) and \( P \).

6. **Final Inequality**:
- From the previous steps, we can derive that:
\[ P^4 \leq 256A \]
- This means that for any triangle, the fourth power of the perimeter is less than or equal to a constant multiplied by the area.

### Conclusion:
The inequalities derived from the properties of triangles provide valuable insights into the relationships between the sides, perimeter, and area. The inequalities we have established are fundamental in triangle geometry.