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Concept

Tangent

A line that touches a curve at a point without crossing it there, pointing in the curve's own direction. For a circle it is perpendicular to the radius, and the square of its length from an outside point is that point's power.

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

A line under every point of x². The curve x² with 4 tangent lines drawn, each extended across the whole interval and each staying below the curve throughout.

A line under every point

The chord above the curve is one definition of convexity. There is a second — a line under the curve at every point, staying under everywhere — and it is the one that turns a statement about a derivative at a point into a statement about the whole function.

analysis · Convexity 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

Named alongside it

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

CircleConvexityCyclic-quadrilateralDerivativeGeometric meanGradientInequalityInscribed angleLocal minimumOptimisationPower of a pointSimilar triangles

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