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Another Conic?

Below, line p is the perpendicular bisector of the segment with endpoints C and D. If you'd like, you can move points C and D around.

Be sure to UNCHECK the checkbox in the lower left corner. Then slowly drag point D around the circle. What do you notice?

Drag the expressions (in the upper right corner) into the explanation below so that it makes logical sense. Refer to the app above as you do.