Radius of Intersecting Circles: Arc SPX Length (3/4)*\pi

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Two circles with equal radii intersect such that the outermost point of either circle passes through the other's center, resulting in an arc SPX with a length of (3/4)*π. To determine the radius of the circles, the central angle SX is crucial, but its identification is unclear. A right triangle can be formed using points S, X, and the midpoint of OS, leveraging symmetry to analyze the angles and relationships. The radii SO and SX are equal, as are OX and SX, leading to further exploration of triangle SOX's properties. Understanding these geometric relationships is essential for solving the problem.
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Two circles with same radius intersect, so that the outermost point of either circle goes through the other's center. Arc SPX has length (3/4)*\pi (see attached image.)

What is the radius of either circle?

I think I could solve this problem if I knew the central angle SX, but I don't know how to identify when they hit (X). Maybe I have to form some special triangle into the image.
 

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Yes, draw some triangles. You want to know what the length of SX is, right? So make it the hypotenuse of a right triangle with vertices S,X, and the midpoint of OS. (Has to be a right triangle by symmetry). Then figure out what kind of special triangle it is and what its angles are, etc.
 
Join OS and OX.

SO and SX are the radii of one circle and OX is the radius of the other circle. Since the two circles have the same radii, OX = SX = SO.
So the triangle SOX is ...? and angle SOX = ...?
 

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