Problem 18: Squares of Chord Segments

Problem 18

Let AB and CD be two perpendicular chords of a circle, intersecting at O. Show that:

AO² + BO² + CO² + DO² = 4R², where R is the radius of the circle

Here’s a simulation that demonstrates the problem.

Try dragging the points A and O. Is AO² + BO² + CO² + DO² always equal to 4R²? How can we prove the same?

Solution

Right angles and squares of lengths. Whenever you see them together, think of Pythagoras’ theorem.

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