유클리드 기본원론 제안 35. 같은 밑변과 한 평행선에 위변을 가진 평행사변형의 면적은 같다

제안 35. 같은 밑변과 위변이 같은 평행선에 있는 평행사변형은 같은 면적을 같는다.

Proposition 35 Parallelograms which are on the same base and in the same parallels equal one another.

 

 

Let ABCD and EBCF be parallelograms on the same base BC and in the same parallels AF and BC.

I say that ABCD equals the parallelogram EBCF.

Since ABCD is a parallelogram, therefore AD equals BC.

For the same reason EF equals BC, so that AD also equals EF. And DE is common, therefore the whole AE equals the whole DF.

But AB also equals DC. Therefore the two sides EA and AB equal the two sides FD and DC respectively, and the angle FDC equals the angle EAB, the exterior equals the interior. Therefore the base EB equals the base FC, and the triangle EAB equals the triangle FDC.

Subtract DGE from each. Then the trapezium ABGD which remains equals the trapezium EGCF which remains.

Add the triangle GBC to each. Then the whole parallelogram ABCD equals the whole parallelogram EBCF.

Therefore parallelograms which are on the same base and in the same parallels equal one another.

 

 

### 덧붙이는 말 ###

 

 

 

 

 

 

 

 

 

 

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