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Concept

Power of a point

The product of the two distances from a point to a circle along any line through the point, which is the same for every such line. It is positive outside the circle, negative inside and zero on it, and equals the squared tangent length outside.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

Every chord through the point cuts into pieces whose product is 16.00. A circle with a point inside it and four chords drawn through the point, each labelled with the lengths of its two pieces, beside four rectangles whose sides are those pieces and whose areas are all equal.

One number for every chord through a point

Draw any line through a point and let it cut a circle twice. The two distances from the point to the circle multiply to the same number whichever line is drawn — inside, outside, or grazing as a tangent. The number belongs to the point, and the reason it does not depend on the line is the inscribed angle: two chords through a point cut out two triangles with the same angles.

geometry · Inscribed angle The butterfly: wings that cut a chord equally. A circle with chord PQ, its midpoint M, two chords through M, and the wings AD and BC meeting PQ at X and Y, each 0.476 from M.

Wings that cut a chord equally

Take a chord of a circle and its midpoint. Draw any two more chords through the midpoint, join their ends crosswise, and the two crossing lines — the butterfly's wings — cut the first chord at equal distances from the middle. The proof is the inscribed angle and the power of a point working together; move the point off the middle and what survives is a law about reciprocals; replace the circle by any conic and the theorem does not notice.

geometry · Inscribed angle

Named alongside it

The objects these essays reach for when they reach for this one.

CircleInscribed angleSimilar trianglesChordConicCyclic-quadrilateralGeometric meanInvariantProjective geometrySineSymmetryTangent

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