another question... circle geometry (1 Viewer)

[mojako]

Ok, here is the question:
Prove that: the difference of the squares of the direct and indirect common tangents of two non-overlapping circles is the product of the two diameters.
When I attempted that question, I got to the point where "it is not true..."
Thank you.

 

[Affinity]

Let CE=ED = x
let EA = EB = y

|AD^2 - BE^2| = |(x+y)^2 - (x-y)^2| = 4xy

But CPE is similar to AEO (why? consider angle sum at E)

so EA/OA = PC/EC

EC*EA = OA * PC = product of radii

xy = product of radii

4xy = product of diameter

 

[mojako]

Wow, thanks Affinity
Very clever observation...

BTw, in my first post I said I came to the point where "it is not true". Here is what I did (see the picture.. I handwrite it). Maybe if you have some spare time you could have a look at it... and give some comments.
*only* if you have the time (it's not that important for me)
Thanks

 

[Affinity]

how do you know it doesn't work?
You can't substitute arbitary values for d1, d2
they depend on each other.

from your proof consider triangles OIE and ZJE.

OE/OI = ZE/ZJ

(r+d1)/r = (s + d2)/s
1 + d1/r = 1 + d2/s
s*d1 = r*d2

and the result needed follows

 

[mojako]

I think I need to learn a lot from you
Thanks.