Proof: By Euclid
(related to Corollary: 6.08: Geometric Mean Theorem)
- By hypothesis, the triangle with with the sides \(a\), \(b\), \(c\) is a right-angled triangle, with the with the right angle between the sides \(a\) and \(b\).
- By proposition 6.08, if, in a right-angled triangle, a segment \(h\) is drawn from the right angle perpendicular to the base \(c\), cutting it into pieces \(p\) and \(q\), then the triangles with the sides \(b\), \(h\), \(p\) and \(a\),\(q\) \(h\) are similar to the whole (triangle) with the sides \(c\), \(a\), \(b\), and to one another.
- Therefore, the side lengths are proportional:
\[\frac hp=\frac qh\]
- Equivalently by proposition 7.19, \(h^2=pq\).
- Thus, \(h=\sqrt{pq}\).
- Thus, \(h\) is the geometric mean of \(p\) and \(q\).
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