Question 407117
Yes, the altitude divides the hypotenuse into two segments:

{{{drawing(200, 200, -10, 10, -1, 10,


triangle(-9, 0, 9, 0, -5, sqrt(46)),
line(-5, sqrt(46), -5, 0),
locate(-9, 0, A)
locate(-5, sqrt(46), B),
locate(9, 0, C),
locate(-5, 0, D)

)
}}}


I think you want to show that the geometric mean of the two segments equals the length of the altitude, which is a famous theorem.


The proof comes from the fact that triangles ABD and CBD are similar, which I've already shown. Therefore, we can let {{{AB = x}}}, {{{BD = xy}}}, and {{{DC = xy^2}}}. The geometric mean of AB and DC is given by


{{{sqrt(x*(xy^2)) = xy}}} which is the same as BD, so we're done.