Proposition 21

Theorem

If two lines (BD , CD ) are drawn to a point (D) within a triangle (△ABC) from the endpoints of its base (BC ), then their sum is less than the sum of the remaining sides (ￜBD ￜ + ￜCD ￜ < ￜBA ￜ + ￜCA ￜ), but they contain a bigger angle (∠BDC > ∠BAC).

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Commentary

Let △ABC be a given triangle, and let D be any point inside △ABC.

Connect D to both ends of the base BC  so that line BD  and line CD  are constructed.

Then the sum of two sides BD  and CD  of △DBC is less than the sum of two sides BA  and CA  of △ABC. Furthermore, ∠BDC (contained by BD  and CD ) is greater than ∠BAC (contained by BA  and CA ).

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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