HERON'S FORMULA: A Geometric Proof

Question

I've found a geometric proof of Heron's formula here LINK and I have trounble with undestanding why three parts of $AY$ segment are equal $s-b$, $s-a$ and $s-c$. How to derive it?

Answer 1

Consider the three points of tangency between the incircle and $\Delta ABC$. We already have $X$ labeled, let $W$ be the tangent point on $BA$ and $V$ the one on $BC$.

We will repeatedly use the fact that the two tangents drawn from a fixed point to a fixed circle have the same length. Thus, $AX=AW$ and so on. We denote: $$AX=\lambda=AM\;\;BW=\mu=BV\;\;CV=\nu=CX$$

Note: you are trying to show that $\lambda = s-a$ and so on. We'll just show that one (the others are similar).

Inspection then shows that $$\lambda+\nu = b$$ $$\lambda+\mu=c$$ $$\mu+\nu=a$$ Adding the first two gives $$2\lambda +(\mu+\nu)=b+c\implies 2\lambda =b+c-a$$ from which your desired result follows at once.