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To find the Center of a Circle.
IF you would find the Center of the Circle AEBD, draw the Line AB, and divide the same in the middle, in the Point C; at which Point erect a Perpendicular ED, which you shall di∣vide also equally in the Point F. This Point F shall be the Center of the Circle; for if it be not, imagine, if you please,
that the Point G is the Center: draw the Lines GA, GB, GC.
Demonstration. If the Point G were the Center, the Triangles GAC, GBC, would have the sides GA, GB, equal by the definition of a Circle: AC, CB, are equal to the Line AB, having been divided in the middle, in the Point C. And CG being common, the Angles GCB, GCA, would then be equal (by the 8th. of the 1st.) and CG would be then a Perpendicular, and not CD; which would be contrary to the Hy∣pothesis. Therefore the Center cannot be out of the Line CD. I further add, that it must be in the Point F, which divideth the same into two equal Parts; otherwise the Lines drawn from the Center to the Circumference would not be equal.
Corollary. The Center of a Circle is in a Line which divideth another Line in the middle, and that perpendicularly.
THis first Proposition is necessary to demonstrate those which follow.