Proof that Perimeter of a Triangle is Greater than the Sum of its Altitudes

1. Perimeter of a Triangle: The perimeter of a triangle is the sum of the lengths of its three sides. Let the sides of the triangle be a, b, and c.

2. Altitudes of a Triangle: The altitudes of a triangle are the perpendicular segments from each vertex to the opposite side. Let the altitudes of the triangle be h₁, h₂, and h₃.

3. Relationship between Perimeter and Altitudes: The sum of the altitudes of a triangle is always less than the perimeter of the triangle.

4. Proof:
- Consider the altitude h₁ from vertex A to side BC. The length of h₁ is less than the length of side a.
- Similarly, h₂ is less than b and h₃ is less than c.
- Therefore, h₁ + h₂ + h₃ < a="" +="" b="" +="" />

5. Conclusion: The sum of the altitudes of a triangle is always less than the perimeter of the triangle.

6. Example:
- Consider a triangle with sides of lengths 5, 12, and 13.
- The altitudes of this triangle are 5, 2.4, and 1.85.
- The perimeter of the triangle is 30.
- The sum of the altitudes is 9.25.
- 30 > 9.25, which confirms that the perimeter is greater than the sum of the altitudes.

Therefore, it can be concluded that the perimeter of a triangle is always greater than the sum of its altitudes.