Every Parallelogram Is a Rectangle

Given parallelogram ABCD we denote two of its opposite sides a and the other b. The diagonals are d₁ and d₂.

Let denote H(p, q, r) Heron's formula for a triangle with sides p, q, r.

Consider triangles ABC and DAB; there sides are a, b, d₁ and a, b, d₂, respectively. The equality of the areas could be expressed in terms of Heron's formula:

H(a, b, d₁) = H(a, b, d₂),

from which we conclude that d₁ = d₂. However, a parallelogram with equal diagonals is necessarily a rectangle.

This conundrum is due to Vladimir Nikolin, an elementary school teacher from Serbia.

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