Question

please slove it in 5 minutes please prove the herons formula in very simple way​

Answer 1

Step-by-step explanation:

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.

Note: let the angles of the triangle be 2α, 2β, 2γ so that α + β + γ = 90°. The identity xyz = r²(x + y + z) is equivalent to the following trigonometric formula:

cotα + cotβ + cotγ = cotα cotβ cotγ,

where "cot" denotes the standard cotangent function.