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Heron's formula is a magical tool for calculating the area of a triangle when you know the lengths of all three sides. This formula is especially handy because you don't need to know any angles. Here's the formula:\[ A = \sqrt{s(s-a)(s-b)(s-c)} \]where \( A \) is the area of the triangle, and \( s \) is the semi-perimeter of the triangle. The variables \( a \), \( b \), and \( c \) represent the lengths of the sides of the triangle. This formula lets you easily find out if the area is an integer, like in a Heron triple. If it turns out to be an integer, it means the set of numbers can be a candidate for a Heron triple.Using Heron's formula involves a bit of calculation:

• First, find the semi-perimeter \( s \) (we'll explain how in the next section).
• Next, use \( s \) to calculate \( A \).
• Finally, check if \( A \) is an integer.

In our examples, for the triangle (5,6,7), we found the area is \( 6 \sqrt{6} \), not an integer. So (5,6,7) is not a Heron triple. For (13,14,15), the area is 84, an integer! Therefore, (13,14,15) forms a Heron triple.

The triangle inequality is a fundamental rule in geometry that helps us verify if three lengths can make a triangle. This rule states that for three lengths to form a triangle, the sum of the lengths of any two sides must be greater than the length of the third side. Let's check what this means with examples:

• For the triangle with sides (5, 6, 7):
  - 5 + 6 = 11, which is greater than 7.
  - 5 + 7 = 12, which is greater than 6.
  - 6 + 7 = 13, which is greater than 5.
  All conditions are satisfied, so (5, 6, 7) can form a triangle.
• Similarly, for the sides (13, 14, 15):
  - 13 + 14 = 27 is greater than 15.
  - 13 + 15 = 28 is greater than 14.
  - 14 + 15 = 29 is greater than 13.
  Hence, these lengths satisfy the triangle inequality, too.

This step is vital because even if side lengths might seem like they could form a triangle, failing even one of these inequalities means they actually can't.

The semi-perimeter of a triangle is simply half of its perimeter. It's an essential part of Heron's formula and makes working with triangles easier. You calculate it with the formula:\[ s = \frac{a + b + c}{2} \]Here, \( s \) is the semi-perimeter, and \( a \), \( b \), and \( c \) are the side lengths of the triangle.Here's how we use it:

• For the triangle (5, 6, 7):
  - \( s = \frac{5 + 6 + 7}{2} = 9 \).
• For the triangle (13, 14, 15):
  - \( s = \frac{13 + 14 + 15}{2} = 21 \).

The semi-perimeter helps break down the triangle's properties and is a stepping stone in using Heron's formula. It offers us a middle ground between simple side lengths and complex area calculations, neatly condensing all side length information into one handy value. This becomes crucial when calculating the area and is a quick plug in Heron's formula for figuring out Heron triples.