S2E39: Heron's formula Divided And Conquered

Описание: In this episode we use the "Divide And Conquer" problem-solving approach in order to generate two distinct proofs of the Heron's formula for the area of an arbitrary planar, Euclidean, triangle.

The technical essence of the first such proof amounts to a relationship between the magnitudes of two angles generated by the InCircle and by an ExCircle of the parent triangle. This proof opens this episode starting at 3:33 and then hops along 4:39, 6:03, 9:29 and 20:22.

The essence of the second proof of the Heron's formula, which begins at 1:18:29, amounts to a combination of an observation and a theorem. The observation that makes this proof tick is the fact that 6 certain triangles that partition the parent triangle generate certain angles that sum to 2 pi radians. The theorem that takes that observation all the way to the bank states that if 3 real numbers sum to pi radians then the sum of the values of the corresponding trigonometric functions of tangent is equal to the product of these values.

But do not lose the forest for the trees: all the while, the clothes line on which all these wonderful technical facts and manipulation hang is none other than the "Divide And Conquer" problem-solving approach.