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In high school mathematics, where is the center of the circumscribed circle of a triangular pyramid with a right triangle or an equilateral triangle at the bottom?
I think you may not be qualified, because a surface can only determine one section of the receiving ball, and the vertex will still affect the position.

If the bottom is a right triangle, then the circumscribed circle of the bottom is at the midpoint of the hypotenuse, and the circumscribed spherical center must be on a straight line perpendicular to the bottom with the midpoint of the hypotenuse as the vertical center.

The bottom is a regular triangle, so the center of the circumscribed circle of the bottom is at the center of the regular triangle, so the center of the circumscribed ball must be on a straight line perpendicular to the bottom with the center of the bottom as the vertical center.