In a right triangle, c is the length of the hypotenuse, a and b are the length of the two other sides, and d is the length of the diameter of the inscribed circle. Prove that a = b = c + d.

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In general, it is not true that `a = b ` and it is always not true that `b = c + d ` because `b lt c . ` Let's determine what equation is true.

Recall any radius drawn from the center of the inscribed circle to the point of tangency is perpendicular to the corresponding side (see the attached image). Because the angle between `a ` and `b ` is also right, there is a square at the left lower corner of the picture.

This way, points of tangency divide the side `a ` into segments of the lengths `d` and `a - d ` and the side `b ` into `d ` and `b - d .`

The corresponding segments on the side `c ` have the same lengths `a - d ` and `b - d . ` And the sum of these lengths is obviously `c , ` so we obtain `a - d + b - d = c .`

It can be also written as `a + b = c + 2 r .`

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