Concept

Converse

The statement obtained from an if-then statement by swapping its hypothesis and its conclusion. A theorem's converse is a separate claim that may be true or false, and proving it is often what turns a property into a test.

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

A knotted rope pulled into a 3-4-5 triangle. A closed loop of rope carrying 12 equally spaced knots, held at three of them so the sides are 3, 4 and 5 knots long; the angle between the two shorter sides is 90.0 degrees.

The rope that squares a corner

The theorem turns two sides into a third. Run it backwards and it turns three lengths into a right angle — which is a different statement, needs its own proof, and is the only one of the two that has ever been used to build anything.

geometry · Pythagoras

Two ways to cut a hexagon, one total of radii. A cyclic hexagon triangulated two ways with the incircles drawn; the inradii sum to 1.1622 both times.

A total hung in a temple

Put a polygon's corners on a circle, cut it into triangles, and add up the radii of the circles inscribed in the triangles. Cut it a different way and the total is the same — for all fourteen ways of cutting a hexagon, to every digit. The fact was painted on a wooden tablet in a Japanese temple around 1800, and the reason for it is a theorem about one triangle and the distances from its circumcentre to its sides.

geometry · Inscribed angle

Named alongside it

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

Catalan numbers Circumcircle Congruence Construction Cyclic polygon Exhaustive search Incircle Inscribed angle Invariant Law of cosines Pythagorean theorem Pythagorean triples

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