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Perpendicular Bisectors of Chords

Point A is the centre of the circle. The line connecting points B and C is a chord in the circle. A line that is perpencicular to the chord and passing through the centre of the circle is drawn. Drag the points around to change the chord's length and location. Notice the length of BD and CD (the line segments created by intersecting the chord). What is ALWAYS true about where the perpendicular line intersects the chord?

Chord Bisector Demo