Rule

Geometric Mean Leg Theorem

Given a right triangle, if the altitude is drawn from the vertex of the right angle to the hypotenuse, then the measure of each leg of the triangle is the geometric mean between the length of the hypotenuse and the length of the segment formed on the hypotenuse adjacent to the leg.

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Based on the diagram above, the following relations hold true.

$⎩ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎧ \frac{A C}{A D} = \frac{A B}{A C} \frac{C B}{D B} = \frac{A B}{C B} or {A C^{2} = A D \cdot A B C B^{2} = D B \cdot A B$

Proof

According to the Right Triangle Similarity Theorem, the two triangles formed by the altitude $\overline{C D}$ are similar to $△ A B C .$ $△ A B C \sim △ A C D △ A B C \sim △ C B D$ Then, by definition of similar triangles, the length of corresponding sides are proportional.

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

Applying the Properties of Equality, the proportion above can be rewritten without fractions.

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