Proposition: 6.05: Triangles with Proportional Sides are Similar

(Proposition 5 from Book 6 of Euclid's “Elements”)

If two triangles have proportional sides then the triangles will be equiangular, and will have the angles which corresponding sides subtend equal. * Let $ABC$ and $DEF$ be two triangles having proportional sides, (so that) as $AB$ (is) to $BC$, so $DE$ (is) to $EF$, and as $BC$ (is) to $CA$, so $EF$ (is) to $FD$, and, further, as $BA$ (is) to $AC$, so $ED$ (is) to $DF$. * I say that triangle $ABC$ is equiangular to triangle $DEF$, and (that the triangles) will have the angles which corresponding sides subtend equal. * (That is), (angle) $ABC$ (equal) to $DEF$, $BCA$ to $EFD$, and, further, $BAC$ to $EDF$.

fig05e

Modern Formulation

Similar triangles are equiangular (this is the converse of Prop. 6.04).

Proofs: 1

Definitions: 1
Proofs: 2


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References

Adapted from (subject to copyright, with kind permission)

  1. Fitzpatrick, Richard: Euclid's "Elements of Geometry"

Adapted from CC BY-SA 3.0 Sources:

  1. Prime.mover and others: "Pr∞fWiki", https://proofwiki.org/wiki/Main_Page, 2016