Proposition 42

Construction

To construct a parallelogram equal to a given triangle, with one angle equal to a given angle.

---

Construction Steps

Let △ABC be the given triangle and let ∠DEF be the given angle.

Construct the infinite line infinite line CP parallel to AB.

Bisect AB at M.

Choose N on infinite line CP such that ∠BMN = ∠DEF.

Construct a line through B that is parallel to MN and intersects infinite line CP at Q. The parallelogram ◇BMNQ has been constructed equal to the given triangle △ABC, with ∠BMN equal to the given angle ∠DEF.

---

---

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[]=

---

Additional instances

• 
• 
• 
• 
• 

---

Dependency graphs