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Problem 550

AMC 12 late, AIME early

Geometry Difficulty 5.0 Find the answer

A circle intersects the coordinate axes at points $A(a, 0), B(b, 0) C(0, c)$ and $D(0, d)$ . Find the coordinates of its center.

Your answer

Official solution

The center $O(x, y)$ of the given circle is the intersection of the perpendicular bisectors of the chords $A B$ and $C D$ . The equations of these perpendiculars are:

$x=1 / 2(a+b)$ and $y=1 / 2(c+d)$ respectively.

## Answer

$\left(\frac{a+b}{2} ; \frac{c+d}{2}\right)$

Source: NuminaMath-1.5, licensed Apache-2.0. Statement and solution reproduced as published; topic, difficulty and ordering added by this site.