유클리드 기본모음 제안 44. 주어진 삼각형의 면적과 주어진 내각을 갖는 평행사변형의 작도

제안 44. 주어진 한 직선과 두 직선이 이루는 한 각에 대하여, 주어진 삼각형과 같은 면적과 주어진 각과 같은 크기의 내각을 갖는 평행사변형을 작도하기.

Proposition 44 To a given straight line in a given rectilinear angle, to apply a parallelogram equal to a given triangle.

 

 

Let AB be the given straight line, D the given rectilinear angle, and C the given triangle.

It is required to apply a parallelogram equal to the given triangle C to the given straight line AB in an angle equal to D.

Construct the parallelogram BEFG equal to the triangle C in the angle EBG which equals D, and let it be placed so that BE is in a straight line with AB. I.42

Draw FG through to H, and draw AH through A parallel to either BG or EF. Join HB.

Since the straight line HF falls upon the parallels AH and EF, therefore the sum of the angles AHF and HFE equals two right angles. Therefore the sum of the angles BHG and GFE is less than two right angles. And straight lines produced indefinitely from angles less than two right angles meet, therefore HB and FE, when produced, will meet.

Let them be produced and meet at K. Draw KL through the point K parallel to either EA or FH. Produce HA and GB to the points L and M.

Then HLKF is a parallelogram, HK is its diameter, and AG and ME are parallelograms, and LB and BF are the so-called complements about HK. Therefore LB equals BF.

But BF equals the triangle C, therefore LB also equals C.

Since the angle GBE equals the angle ABM, while the angle GBE equals D, therefore the angle ABM also equals the angle D.

Therefore the parallelogram LB equal to the given triangle C has been applied to the given straight line AB, in the angle ABM which equals D.

 

 

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