Proposition 48

Theorem

Alternate name(s): Pythagorean theorem converse.

If the area of the square on one side (AB) of a triangle (△ABC) is equal to the sum of the areas of the squares on the remaining sides (AC, BC), then the angle (∠ACB) opposite to that side is a right angle.

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Commentary

Let △ABC be a given triangle.

On each side of △ABC, let a square be constructed.

If the area of the square on one side AB (◇ABFG) is the sum of the areas of the two squares on the other two sides AC and BC (◇ACHK and ◇BCDE), then the angle ∠ACB opposite to side AB is a right angle.

This geometric relationship between the areas of the constructed squares is often expressed algebraically as a² + b² = c², meaning ￜACￜ² + ￜBCￜ² = ￜABￜ².

This proposition is the converse of the Pythagorean theorem appearing in Book 1 Proposition 47.

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Computable version

In[1]:=

GeometricScene[{{A.,B.,C.,O.,D.,E.,F.,G.},{}},​​{GeometricStep[{CircleThrough[{A.,B.,C.},O.],Line[{A.,O.,C.}],Line[{D.,E.}],EuclideanDistance[D.,E.]≤EuclideanDistance[A.,C.]}],​​GeometricStep[{F.∈Line[{A.,C.}],EuclideanDistance[A.,F.]EuclideanDistance[D.,E.]}],​​GeometricStep[{GeometricAssertion[{CircleThrough[{F.},A.],CircleThrough[{A.,B.,C.},O.]},{"Concurrent",G.}]}],​​GeometricStep[{Line[{A.,G.}]}]},​​{EuclideanDistance[A.,G.]EuclideanDistance[D.,E.]}​​]//RandomInstance

Out[]=

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