Let's begin by considering a right triangle ABC, and let's draw the altitude from the right angle to the hypotenuse. Notice that the legs of the triangle are also altitudes.
Next, we will separate the three triangles formed above.
By applying the Angle-Angle (AA) Similarity Theorem, we get that △ ABC ~ △ ACD and △ ABC ~ △ CBD. Then, by the Transitive Property of Similarity, we have that the two smaller triangles are similar.
△ ACD ~ △ CBD
By definition of similar triangles, we have that the corresponding side lengths are proportional.
AC/CB = CD/BD = AD/CD
Notice that the right-hand equation implies that CD is the geometric mean of AD and BD. Then, we can write a first relation between altitudes and geometric means of right triangles.
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of the altitude is the geometric mean of the lengths of the two segments of the hypotenuse.
Next, let's write the proportions corresponding to the other two triangle similarities.
Triangle Similarity
Proportions
△ ABC ~ △ ACD
AB/AC = AC/AD = BC/CD
△ ABC ~ △ CBD
AB/CB = BC/BD = AC/CD
From the table above, we see that AC is the geometric mean of AB and AD. Also, BC is the geometric mean of AB and BD. This leads us to a second relation between altitudes and geometric means of a right triangle.
In a right triangle, the altitude from the right angle to the hypotenuse divides the hypotenuse into two segments. The length of each leg of the right triangle is the geometric mean of the lengths of the hypotenuse and the segment of the hypotenuse that is adjacent to the leg.
