Question

verification of herons formula​

Answer 1

Answer:

Trigonometry/Proof: Heron's Formula. We know that a triangle with sides 3,4 and 5 is a right triangle. Two such triangles would make a rectangle with sides 3 and 4, so its area is 3 ⋅ 4 2 = 6 {\displaystyle {\frac {3\cdot 4}{2}}=6} .

Answer 2

In ΔABC, the lengths of the segments from vertices to the points of tangency of the incircle are found to be

x = s - a, y = s - b, z = s - c,

so that Heron's formula can be also written as S² = sxyz.

Let r be the inradius of ΔABC. The rearrangement of the six triangles of the dissection as done at the bottom of the applet, shows immediately that S = rs.

Let I be the incenter and denote w = AI. From the diagram in the right portion of the applet,

xyz = r²(x + y + z) = r²s.

It then follows that sxyz = r²s² = S², which completes the proof.