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Learn about packing shapes neatly into others, and the resulting geometric properties.

In the right triangle above and altitude is drawn from the right angle to the hypotenuse. Circles are inscribed within each of the smaller triangles.

What is the distance between the centers of these circles?

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The 5 circles have the same size. If the side of the large square is 1, what is the radius of each circle?

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Triangle \(ABC\) is such that \(AB = 13, BC = 14,\) and \(CA = 15.\) A point \(D\) on \(BC\) is placed such that \(AB + BD = AC + CD.\) Let \(X\) be the intersection of \(AD\) with the incircle of \(ABC\) closest to \(A.\) Find the length of \(BX.\)

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Find the value of \(x\).

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