Geometric Mean Leg Theorem

According to the Right Triangle Similarity Theorem, the two triangles formed by the altitude

\overline{C D}

are similar to

△ A B C .

△ A B C \sim △ A C D △ A B C \sim △ C B D

This means that the corresponding parts of the triangles can be identified.

$△ A B C \sim △ A C D$	$△ A B C \sim △ C B D$
$\overline{A B} and \overline{A C}$ $\overline{A C} and \overline{A D}$ $\overline{C B} and \overline{C D}$	$\overline{A B} and \overline{C B}$ $\overline{A C} and \overline{C D}$ $\overline{C B} and \overline{D B}$

Then, by definition of similar triangles, the lengths of corresponding sides are proportional.

$△ A B C \sim △ A C D$	$△ A B C \sim △ C B D$
$\frac{A C}{A D} = \frac{A B}{A C}$	$\frac{C B}{D B} = \frac{A B}{C B}$

Note that for any two pairs of corresponding sides a similar proportion can be obtained. Now, applying the Properties of Equality, the proportion can be rewritten without fractions.

{A C^{2} = A D \cdot A B C B^{2} = D B \cdot A B

Geometric Mean Leg Theorem

Proof

Recommended exercises