Proposition 8

Theorem

Alternate name(s): SSS theorem.

If two triangles (△ABC, △DEF) have two sides of one respectively equal to two sides of the other (ￜAB ￜ = ￜDE ￜ, ￜAC ￜ = ￜDF ￜ), and have also the base of one equal to the base of the other (ￜBC ￜ = ￜEF ￜ), then the two triangles are congruent.

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Commentary

Let △ABC and △DEF be two given triangles, with all three pairs of corresponding sides (or as Euclid said, two pairs of sides and a pair of bases) being equal.

These two triangles are said to be congruent, and the three pairs of corresponding angles are equal.

This proposition is known as the SSS (or side-side-side) rule for triangle congruence.

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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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