Proposition 12
Theorem
In any triangle △ABC , with ∠ACB > 90° , extend BC such that it meets the perpendicular AD from the opposite angle at point D . Then ￜABￜ2 = ￜBCￜ2 + ￜCAￜ2 + 2⋅ￜBCￜ⋅ￜCDￜ .
Commentary
1. Let △ABC be the given obtuse triangle with C being the obtuse angle.
2. ExtendBC such that it meets the perpendicular from vertex A at a point D .
3. Two right triangles are constructed:△ABD and △ACD .
4. Apply the Pythagorean Theorem to the right triangles so: in△ABD , ￜABￜ2 = ￜBDￜ2 + ￜADￜ2 , and in △ACD , ￜCAￜ2 = ￜCDￜ2 + ￜADￜ2 . Using these equations, and the fact that ￜBCￜ + ￜCDￜ = ￜBDￜ , the following equation is obtained: ￜABￜ2 = ￜBCￜ2 + ￜCAￜ2 + 2⋅ￜBCￜ⋅ￜCDￜ .
5. This algebraic relationship is similar to the next proposition, Book 2 Proposition 13, and is a geometric version of the law of cosines in trigonometry.
6. This proposition is a variation of the Pythagorean Theorem (Book 1 Proposition 47) which applies to obtuse triangles.
2. Extend
3. Two right triangles are constructed:
4. Apply the Pythagorean Theorem to the right triangles so: in
5. This algebraic relationship is similar to the next proposition, Book 2 Proposition 13, and is a geometric version of the law of cosines in trigonometry.
6. This proposition is a variation of the Pythagorean Theorem (Book 1 Proposition 47) which applies to obtuse triangles.
Original statement
ἐν τοῖς ἀμβλυγωνίοις τριγώνοις τὸ ἀπὸ τῆς τὴν ἀμβλϵῖαν γωνίαν ὑποτϵινούσης πλϵυρᾶς τϵτράγωνον μϵῖζόν ἐστι τῶν ἀπὸ τῶν τὴν ἀμβλϵῖαν γωνίαν πϵριϵχουσῶν πλϵυρῶν τϵτραγώνων τῷ πϵριϵχομένῳ δὶς ὑπό τϵ μιᾶς τῶν πϵρὶ τὴν ἀμβλϵῖαν γωνίαν, ἐϕ᾽ ἣν ἡ κάθϵτος πίπτϵι, καὶ τῆς ἀπολαμβανομένης ἐκτὸς ὑπὸ τῆς καθέτου πρὸς τῇ ἀμβλϵίᾳ γωνίᾳ.
English translation
In obtuse-angled triangles the square on the side subtending the obtuse angle is greater than the squares on the sides containing the obtuse angle by twice the rectangle contained by one of the sides about the obtuse angle, namely that on which the perpendicular falls, and the straight line cut off outside by the perpendicular towards the obtuse angle.
