Proposition 32

Theorem

Alternate name(s): triangle postulate.

If any side (AB) of a triangle (△ABC) is extended (to D), then the exterior angle (∠CBD) is equal to the sum of the two non-adjacent interior angles (∠BAC, ∠BCA), and the sum of the three internal angles is equal to two right angles.

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Commentary

Let △ABC be a given triangle. Extend the side AB to a point D.

Then the exterior angle ∠CBD is equal to the sum of the two non-adjacent interior angles ∠BAC and ∠BCA. Moreover, the three interior angles add up to two right angles (180 degrees).

This proposition is also known as the Triangle Postulate and is a stronger version of Book 1 Proposition 16.

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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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