Prove on circle theorems help

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so I have proved the circle theorem, the one that states the central angle is always twice the circum angle and that any circum angle that subtends the same arc always have the same measure, but I proved it case by case. Is there a better way to prove it?
and my 2nd question:
is there a way to prove that a tangent line to a circle always form a right angle with the radius without using coordinate geometry?

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