Proposition 18

Theorem

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

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Commentary

Let △ABC be a given triangle, with side AC  longer than side AB .

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

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

The next proposition, Book 1 Proposition 19, is the converse of this one.

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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