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Two circles are congruent circles if and only if they have the same radius.
Based on the above diagram, the theorem can be written as follows.
⊙ A ≅ ⊙ B ⇔ r_1=r_2
Consider two congruent circles ⊙ A and ⊙ B and a point on each one.
Because ⊙ A ≅ ⊙ B, the distance from the center to a point on the circle is the same for both circles. Therefore, AC≅ BD, which implies that both circles have the same radius. That is, r_1=r_2.
It has been proved that if two circles are congruent, then they have the same radius.
Consider now two circles with the same radius.
Next, ⊙ B can be translated so that point B is mapped onto point A. The image of ⊙ B is ⊙ B', which is a circle centered at A. Since the circles have the same radius, this translation maps ⊙ B onto ⊙ A.
Because a rigid motion maps one circle onto the other, it is concluded that both circles are congruent.