SOLUTION: if y is the mean proportional between x and z ; show that xy + yz is the mean proportional between x² + y² and y² + z²

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Question 1121372: if y is the mean proportional between x and z ; show that xy + yz is the mean proportional between x² + y² and y² + z²
Found 2 solutions by ankor@dixie-net.com, greenestamps:
Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
You can put this solution on YOUR website!
if y is the mean proportional between x and z ;
We know:
x%2Fy = y%2Fz; y+=+sqrt%28xz%29
:
show that xy + yz is the mean proportional between x² + y² and y² + z²
%28x%5E2%2By%5E2%29%2F%28xy%2Bzy%29 = %28xy%2Bzy%29%2F%28y%5E2%2Bz%5E2%29
The easiest to show this is to give some values to x & z
x=2; z=18; then y = sqrt%282%2A18%29 = 6
:
%284%2B36%29%2F%2812%2B108%29 = %2812%2B108%29%2F%2836%2B324%29
40%2F120 = 120%2F360

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!
We are given that

x%2Fy+=+y%2Fz

or

(1) y%5E2+=+xz

Then we are to show that

(2) %28x%5E2%2By%5E2%29%2F%28xy%2Byz%29+=+%28xy%2Byz%29%2F%28y%5E2%2Bz%5E2%29

The algebra for the general case is not terribly difficult....

Equation (2) is equivalent to

%28xy%2Byz%29%5E2+=+%28x%5E2%2By%5E2%29%28y%5E2%2Bz%5E2%29

Then using equation (1) where appropriate....

%28y%28x%2Bz%29%29%5E2+=+%28x%5E2%2By%5E2%29%28y%5E2%2Bz%5E2%29

%28y%5E2%29%28x%2Bz%29%5E2+=+%28x%5E2%2By%5E2%29%28y%5E2%2Bz%5E2%29

%28xz%29%28x%2Bz%29%5E2+=+%28x%5E2%2Bxz%29%28xz%2Bz%5E2%29

%28xz%29%28x%2Bz%29%5E2+=+%28x%29%28x%2Bz%29%28z%29%28x%2Bz%29

%28xz%29%28x%2Bz%29%5E2+=+%28xz%29%28x%2Bz%29%5E2

Done...!