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Actually Δ ADE is not a right triangle. (Unless A is infinitely far away from D and E.)

Edit: thanks TheLoneWolfling, I was looking at the wrong angle.

Err, are you sure about that? It touches at one point, so that makes it a tangent, so it's perpendicular to the circle. I think it it a right angle.

(This is the internet, so I'm sure there will shortly be lots of posts of: Yes it is, no it isn't, and we'll find out for sure).

Wrong.

The angle ADE is a right angle, as AD is tangential to the sphere. Hence, Δ ADE is a right triangle.

I'm assuming your mistake was thinking that it was AED that was the right angle, not ADE.