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Computable Euclid

Proposition 24

Theorem

If two triangles (△ABC, △DEF) have two sides of one respectively equal to two sides of the other (ￜABￜ = ￜDEￜ, ￜACￜ = ￜDFￜ), but have the contained angle of one bigger than the contained angle of the other (∠BAC > ∠EDF), then the base of that which has the bigger angle is longer than the base of the other (ￜBCￜ > ￜEFￜ).

Commentary

  • Let △ABC and △DEF be two given triangles, and let two pairs of sides of these two triangles be equal, namely ￜABￜ = ￜDEￜ and ￜACￜ = ￜDFￜ. Then BC is the base of △ABC and EF is the base of △DEF.
  • Let ∠BAC (contained by AB and AC in △ABC) be bigger than ∠EDF (contained by DE and DF in △DEF).
  • Then the base BC of △ABC (opposite to the bigger angle ∠BAC) is longer than the base EF of △DEF (opposite to the smaller angle ∠EDF).
  • The next proposition, Book 1 Proposition 25, is the converse of this one.
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    Original statement

    ἐὰν δύο τρίγωνα τὰς δύο πλϵυρὰς ταῖς δύο πλϵυραῖς ἴσας ἔχῃ ἑκατέραν ἑκατέρᾳ, τὴν δὲ γωνίαν τῆς γωνίας μϵίζονα ἔχῃ τὴν ὑπὸ τῶν ἴσων ϵὐθϵιῶν πϵριϵχομένην, καὶ τὴν βάσιν τῆς βάσϵως μϵίζονα ἕξϵι.

    English translation

    If two triangles have the two sides equal to two sides respectively, but have the one of the angles contained by the equal straight lines greater than the other, they will also have the base greater than the base.

    Computable version

    Additional instances

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