Proposition 45

Construction

To construct a parallelogram equal to a given polygon, and having an angle equal to a given angle.

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Construction Steps

Let ◇ADCB be the given polygon and let ∠FEG be the given angle.

Join BD.

Construct a parallelogram ◇HJKL that has the same area as triangle △ADB, with ∠LHJ = ∠FEG.

Extend HJ to construct a parallelogram ◇JNMK that has the same area as triangle △BDC, with ∠KJN = ∠FEG. Parallelogram ◇HNML equal to the given polygon ◇ADBC has been constructed with ∠LHN equal to the given angle ∠FEG.

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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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Additional instances

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Dependency graphs