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The circumcircle of triangle ABC goes through the points A, B, and C by the definition of circumcircle.

Let's look at triangle AHM. Point M, which is the circumcenter of ABC, is located along the perpendicular bisector of AB by the Perpendicular Bisector Theroem. Point N, the circumcenter of AHB, must also be located along the perpendicular bisector of AB. Segment HC is parallel to segment NM because both are perpendicular to AB. NH is parallel to MG because both are perpendicular to HJ. Therefore, NHCM is a parallelogram and NH is equal to MC. Since NH is the radius of the circumcircle with center N and MC is the radius of the circumcircle with center M, the radii of these two cricles are equal. This proof can be used in the exact same way to prove that MC=GH and MC=JH