Ramanujan Notebook: rt triangle ABC in which two arcs of circles are centred at A & C

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Ramanujan Notebook: rt triangle ABC in which two arcs of circles are centred at A & C

This result appears in one of Ramanujan's notebooks, I can't see a proof?
Any thoughts?
Take a right-angled triangle ABC in which two arcs of circles are centred at A & C,
Ramanujan gave this result, which i have verified on GGB below and seems to be correct.

DE*DE = 2 AD* EC

 

[yma16]

draw a chord between the two circles

This result appears in one of Ramanujan's notebooks, I can't see a proof?
Any thoughts?
Take a right-angled triangle ABC in which two arcs of circles are centred at A & C,
Ramanujan gave this result, which i have verified on GGB below and seems to be correct.

DE*DE = 2 AD* EC

View attachment 9604

The chord meet the line AC at a point and name it as F. Now, line AC has 5 points ACDEF, which yield 4 segments.
Name them as a,b,c,d. Let the radii be r1 and r2. Then a+b+c=r1, b+c+d=r2, and r1^2+r2^2=(a+b+c+d)^2.
Plug the first two equations into the 3rd, you will get the proof.

 

[apple2357]

WHY do you want to see a proof?

Maybe I should have said help with a proof!