# Find the Fourth Proportional of Three Lengths

### Geometric Construction with the Compass Alone

Let the quantities a, b, c be defined as the lengths of three given segments. Find x such that a/b = c/x.

## Solution

We shall consider three cases:

- c<2a
- c ≥ 2a, b < 2a
- c ≥ 2a, b ≥ 2a

In the case 1, take an arbitrary point O and describe two circles (I and II) with radii a and b, respectively. Pick a point A on the first circle (I) as the center and swing an arc with radius c to find the intersection point B.

Now, with A and B as centers draw two circles of an arbitrary radius

### Proof

Indeed, the triangles ABO and CDO are similar isosceles triangles.

In the second case, consider the proportion

In the third case, use Problem #1 to construct a segment of length na such that

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