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.
We shall consider three cases:
- 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
Indeed, the triangles ABO and CDO are similar isosceles triangles.
In the second case, consider the proportion
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