Proposition 6

Theorem

If in a triangle (△ABC), two angles (∠ABC, ∠ACB) are equal, then the sides (AC , AB ) opposite to them are also equal.

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Commentary

Let △ABC be a given triangle, with ∠ABC and ∠ACB at the base BC  being equal. Then the two sides AC  and AB  opposite to these angles are equal.

Therefore △ABC is an isosceles triangle.

This proposition is the converse of the first part of the previous proposition, Book 1 Proposition 5.

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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

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Additional instances

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Dependency graphs