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I'm trying to figure out a couple things. The main question I have is how to find the area of a circle inscribed inside a quarter circle with a radius of x. The secondary question to that is if the radius of the inner circle drawn to touch the tangent lines of the sides of the quarter circle bisects the sides of the quarter circle. I hope that makes sense...

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1 Answer

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A figure helps a lot. The three segments labeled $r$ are all radii of the small circle. The diagonal of the square is $r\sqrt 2$, so $r(1+\sqrt 2)=x$ No, the tangency points do not bisect the radii of the big circle, they cut them in ratio $1: \sqrt 2$

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