Proposition 7

Theorem

If two triangles (△ACB, △ADB) on the same base (AB ) and on the same side of it have one pair of conterminous sides (AC , AD ) equal to one another, then the other pair of conterminous sides (BC , BD ) must be unequal.

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Commentary

Let AB  be a given line segment and construct triangles △ABC and △ABD on it, with ￜAC ￜ = ￜAD ￜ and distinct vertices C and D on the same side of AB .

The remaining sides of the two triangles, BC  and BD , are of different lengths.

Note that Euclid originally stated this fact in a negative way: you cannot have two different triangles △ABC, △ABD with the vertices C, D on the same side of AB , ￜAC ￜ = ￜAD ￜ and ￜBC ￜ = ￜBD ￜ.

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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