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Drawing and in-circle and proof

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Let r be the radius on the in-circle of tri ABC and a,b,c be the radii of the ex-circles opposite vertices A, B, and C, respectively. Illustrate the fact that 1/r = 1/a + 1/b + 1/c. Also, write a proof of this.

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This solution provides a detailed step by step explanation of the given discrete math problem.

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The incircle and excirles
There are many nice properties of the incirce and excircles of a given triangle. Let first introduce our notations - see Figure 1. The incircle with radius r (inradius) and center I touches the sides BC, CA and AB at X, Y and Z. We know that AY = AZ, BZ = BX and CX = CY. We have accordingly labeled these segments x, y and z, so that:
(please see the attached file)
(We are using the standard notation: a, b, c and p for the sides BC, AC, AB and the semi-perimeter of (please see the attached file))
y + z = ...

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