Euclid Book 13 Proposition 7a
Statement
If three angles of an equilateral pentagon, taken in order, are equal, then the pentagon is equiangular.
Computational Explanation
GeometricSceneA.,B.,C.,D.,E.,{},GeometricAssertionPolygonA.,B.,C.,D.,E.,"Convex","Equilateral",PlanarAngle[B.,A.,E.]PlanarAngle[A.,E.,D.]PlanarAngleE.,D.,C.,GeometricAssertionPolygonA.,B.,C.,D.,E.,"Equiangular"
GeometricAssertionPolygonA.,B.,C.,D.,E.,Equiangular |
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Explanations
For in the equilateral pentagon let, first, three angles taken in order, those at ,,, be equal to one another; I say that the pentagon is equiangular.
For let, , be joined.
Now, since the two sides, are equal to the two sides , respectively, and the angle is equal to the angle , therefore the base is equal to the base , the triangle is equal to the triangle , and the remaining angles will be equal to the remaining angles, namely those which the equal sides subtend, that is, the angle to the angle , and the angle to the angle ; hence the side is also equal to the side .
But the whole was also proved equal to the whole ; therefore the remainder is also equal to the remainder .ut is also equal to . Therefore the two sides , are equal to the two sides , ; and the base is common to them; therefore the angle is equal to the angle .
But the angle was also proved equal to the angle ; therefore the whole angle is also equal to the whole angle .
But, by hypothesis, the angle is equal to the angles at ,; therefore the angle is also equal to the angles at ,. Similarly we can prove that the angle is also equal to the angles at ,,; therefore the pentagon is equiangular.
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For let
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Now, since the two sides
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But the whole
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But the angle
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But, by hypothesis, the angle
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