Heron's Formula/Proof 1

Theorem
Given a triangle $$\triangle ABC$$ with sides $$a$$, $$b$$, and $$c$$ opposite points $$A$$, $$B$$, and $$C$$, respectively.

Let $$s$$ be the semiperimeter, so $$s = \frac{a + b + c}{2}$$.

Then the area $$A$$ of the triangle is given by the formula $$A = \sqrt{s(s - a)(s - b)(s - c)}$$.

Proof
Construct the altitude from $$A$$. Let the length of the altitude be $$h$$ and the foot of the altitude be $$D$$.

Let the distance from $$D$$ to $$B$$ be $$z$$.


 * Heron's Formula.png

Then $$h^2 + (a - z)^2 = b^2$$ and $$h^2 + z^2 = c^2$$ from the Pythagorean Theorem.

By subtracting these two equations, we get $$2az - a^2 = c^2 - b^2$$, which simplifies to $$z = \frac{a^2 + c^2 - b^2}{2a}$$.

Plugging back in and simplifying yields $$h = \sqrt{c^2 - \left(\frac{a^2 + c^2 - b^2}{2a}\right)^2}$$

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