The area of an arbitrary triangle is less than one fourth the square of its semi perimeter. True or False?

Answer

To determine whether the statement "The area of an arbitrary triangle is less than one fourth the square of its semi-perimeter" is true or false, we can follow these steps:

### Step 1: Define the semi-perimeter
The semi-perimeter \( S \) of a triangle with sides \( a \), \( b \), and \( c \) is defined as:
\[ S = \frac{a + b + c}{2} \]

### Step 2: Use Heron's formula for the area
The area \( A \) of a triangle can be calculated using Heron's formula:
\[ A = \sqrt{S(S - a)(S - b)(S - c)} \]

### Step 3: Set up the inequality
We need to check if:
\[ A < \frac{1}{4} S^2 \]
Substituting Heron's formula into the inequality gives us:
\[ \sqrt{S(S - a)(S - b)(S - c)} < \frac{1}{4} S^2 \]

### Step 4: Square both sides
To eliminate the square root, we square both sides of the inequality:
\[ S(S - a)(S - b)(S - c) < \frac{1}{16} S^4 \]

### Step 5: Rearranging the inequality
Rearranging gives us:
\[ 16S(S - a)(S - b)(S - c) < S^4 \]
Assuming \( S > 0 \) (which is true for any triangle), we can divide both sides by \( S \):
\[ 16(S - a)(S - b)(S - c) < S^3 \]

### Step 6: Analyze the expression
The expression \( (S - a)(S - b)(S - c) \) represents the product of three positive terms (since \( S \) is greater than each side of the triangle). Therefore, we need to analyze whether \( 16(S - a)(S - b)(S - c) < S^3 \) holds true for all triangles.

### Step 7: Use the AM-GM inequality
By the Arithmetic Mean-Geometric Mean (AM-GM) inequality:
\[ \frac{(S - a) + (S - b) + (S - c)}{3} \geq \sqrt[3]{(S - a)(S - b)(S - c)} \]
Calculating the left-hand side:
\[ \frac{3S - (a + b + c)}{3} = \frac{3S - 2S}{3} = \frac{S}{3} \]
Thus, we have:
\[ \frac{S}{3} \geq \sqrt[3]{(S - a)(S - b)(S - c)} \]
Cubing both sides gives:
\[ \left(\frac{S}{3}\right)^3 \geq (S - a)(S - b)(S - c) \]
This implies:
\[ \frac{S^3}{27} \geq (S - a)(S - b)(S - c) \]

### Step 8: Final comparison
Multiplying both sides by 16:
\[ \frac{16S^3}{27} \geq 16(S - a)(S - b)(S - c) \]
Now, we need to check if:
\[ \frac{16S^3}{27} < S^3 \]
This is equivalent to checking if:
\[ \frac{16}{27} < 1 \]
which is true.

### Conclusion
Since the inequality holds true, we conclude that:
\[ A < \frac{1}{4} S^2 \]
Thus, the statement "The area of an arbitrary triangle is less than one fourth the square of its semi-perimeter" is **True**.