Proposition 5

Theorem

Alternate name(s): isosceles triangle theorem, Pons Asinorum, the Bridge of Asses, Elefuga.

The angles (∠ABC, ∠ACB) at the base (BC ) of an isosceles triangle (△ABC) are equal to one another, and if the equal sides (AB , AC ) are extended, the external angles (∠FBC, ∠GCB) below the base will be equal.

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Commentary

Let △ABC be a given isosceles triangle with side ￜAB ￜ = ￜAC ￜ. Then the two angles ∠ABC and ∠ACB at the base BC  are equal.

Extend the two equal sides to F and G, respectively. Then the external angles ∠FBC and ∠GCB under the base are also equal.

Euclid defined isosceles triangles as containing exactly two equal sides. The modern definition requires isosceles triangles to have at least two equal sides.

This proposition is known as the Isosceles Triangle Theorem. The following proposition, Book 1 Proposition 6, is the converse of the first part of this proposition.

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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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