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Geometry Problem 81: Triangle Area, Side, Inradius, Circumradius. High School, College, Math Education

Suppose you have a triangle ABC with an incircle centered at I, with inradius r and circumradius R. Let BC be denoted by a. Prove that the area of triangle DIE is given by the formula:  \(S_1 = \dfrac{a\cdot r^2}{4\cdot R}\).

Triangle, Area, Inradius, CIrcunradius

Suppose a triangle ABC you behold,
With incircle at I, inradius r to hold,
And circumradius R, let BC be denoted a,
Prove that area of triangle DIE is no cliche.

With formulas, proofs, and geometric lines,
This calculation, no longer one that confines,
But reveals the beauty of triangle centers,
A world of shapes, geometry adventure.

Oh, r is the inradius, and R, circumradius,
The calculation's elegance, enough to impress,
And through this math, a deeper meaning revealed,
The intricate patterns of shapes unconcealed.

So let us embrace, with wonder and delight,
The secrets geometry brings to light.

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