How to prove that the point of intersection of the midpoints of the inscribed quadrilateral ABCD

| September 5, 2021 | education

How to prove that the point of intersection of the midpoints of the inscribed quadrilateral ABCD will be the center of the circumcircle?

Let ABCD be a quadrilateral inscribed in a circle, and points A, B, C, D are on a circle with center O. Restore the midpoint perpendicular to side AB (perpendicular from the middle of AB). Then all points on this perpendicular will be equally distant from points A and B (according to the property of the midpoint perpendicular).

Let us restore the middle perpendicular to the side BC, and all points on it are equally distant from points B and C.
The point of intersection of two perpendiculars will give an equidistant point O from points A, B, and C, OA = OB = OC.

Since point D is also on a circle, then OD = ОА = ОВ = OC = the radius of the circle circumscribed around ABCD. And we found the center of the circle by finding the point of intersection of the median perpendiculars.



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