Proposition 19

Theorem

If in any triangle (△ABC) one angle is bigger than another (∠ABC > ∠ACB), then the side (AC ) opposite to the bigger angle (∠ABC) is longer than the side (AB ) opposite to the smaller angle (∠ACB).

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Commentary

Let △ABC be a given triangle, with ∠ABC bigger than ∠ACB.

Then the side AC  opposite to the bigger angle ∠ABC is longer than the side AB  opposite to the smaller angle ∠ACB.

This statement can be summed up by saying that, in a triangle, the side opposite the bigger angle is longer.

This proposition is the converse of the previous proposition, Book 1 Proposition 18.

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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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