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Prove that r(1) r(2) + r(2) r(3) + r(3) ...

Prove that `r_(1) r_(2) + r_(2) r_(3) + r_(3) r_(1) = (1)/(4) (a + b + c)^(2)`

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To prove that \( r_1 r_2 + r_2 r_3 + r_3 r_1 = \frac{1}{4} (a + b + c)^2 \), we will follow these steps: ### Step 1: Understand the relationship between the inradii and the area of the triangle The inradii \( r_1, r_2, r_3 \) of a triangle can be expressed in terms of the area \( \Delta \) and the semi-perimeter \( s \) of the triangle. The semi-perimeter \( s \) is given by: \[ s = \frac{a + b + c}{2} \] where \( a, b, c \) are the lengths of the sides of the triangle. ...
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