Proposition 44

Construction

On a given line segment, construct a parallelogram that is equal to a given triangle, and has one of its angles equal to a given angle.

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Construction Steps

Let AB be the given line segment, △CDE be the given triangle and ∠GFH be the given angle.

Construct a parallelogram ◇BJLK that has the same area as triangle △CED, with J on the extension of AB, and ∠KBJ = ∠GFH.

Extend LK to M such that MA is parallel to BK.

Join MB and extend it to intersect the extension of LJ at N.

Let NP be parallel to JB and intersect the extension of MA at P.

Extend KB such that it intersects NP at Q. Parallelogram ◇PQBA equal to the given triangle △CED has been constructed on the given line segment AB, with ∠ABQ equal to the given angle ∠GFH.

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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