Proposition 20

Theorem

Alternate name(s): triangle inequality.

The sum of any two sides (AB , AC ) of a triangle (△ABC) is greater than the third (BC ).

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Commentary

Let △ABC be a given triangle.

Then the sum of two sides AB  and AC  is greater than the remaining side BC .

This proposition can be summed up by saying, "in a triangle, the sum of any two sides is always greater than the remaining side," or "the shortest path between two points is a straight line."

This proposition is known as the triangle inequality.

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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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