If in a triangle the square on one of the sides be equal to the squares on the remaining two sides of the triangle, the angle contained by the remaining two sides of the triangle is right.
若一个三角形一边上的正方形等于另外两边上的正方形之和，则这个三角形在那两边所夹的角是直角。
三角形 ABC 中，若 AB²+AC²=BC²。从 A 作 AD⊥AC、AD=AB（D 在 AC 远离 B 的一侧）；连接 CD。由两边及夹角可证 △ABC≌△ADC，故 ∠BAC=∠DAC=直角。
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For in the triangle ABC let the square on one side BC be equal to the squares on the sides BA, AC; I say that the angle BAC is right. For let AD be drawn from the point A at right angles to the straight line AC, let AD be made equal to BA, and let DC be joined.
另作一个直角三角形，使它的两条直角边分别等于原三角形的两条较短边。
Since DA is equal to AB, the square on DA is also equal to the square on AB. Let the square on AC be added to each; therefore the squares on DA, AC are equal to the squares on BA, AC.
由 euclid-elements/book1-prop-047，这个直角三角形斜边上的正方形等于两直角边上正方形之和。
But the square on DC is equal to the squares on DA, AC, for the angle DAC is right; [I. 47] and the square on BC is equal to the squares on BA, AC, for this is the hypothesis; therefore the square on DC is equal to the square on BC, so that the side DC is also equal to BC. And, since DA is equal to AB, and AC is common, the two sides DA, AC are equal to the two sides BA, AC; and the base DC is equal to the base BC; therefore the angle DAC is equal to the angle BAC.
已知原三角形某边上的正方形也等于同一和，所以两条斜边相等。
[I. 8] But the angle DAC is right; therefore the angle BAC is also right. Therefore etc.
三边分别相等，由 euclid-elements/book1-prop-008，原三角形对应角为直角。