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Proposition 31

In right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle.
Let ABC be a right-angled triangle having the angle BAC right. I say that the figure on BC equals the sum of the similar and similarly described figures on BA and AC.
Draw the perpendicular AD. | I.12
| Then, since in the right-angled triangle ABC, AD has been drawn from the right angle at A perpendicular to the base BC, therefore the triangles DBA and DAC adjoining the perpendicular are similar both to the whole ABC and to one another. | VI.8
And, since ABC is similar to DBA, therefore BC is to BA as BA is to BD. | VI.Def.1 |
And, since three straight lines are proportional, the first is to the third as the figure on the first is to the similar and similarly described figure on the second. | VI.19,Cor
Therefore BC is to BD as the figure on BC is to the similar and similarly described figure on BA.
For the same reason also, BC is to CD as the figure on BC is to that on CA, so that, in addition, BC is to the sum of BD and DC as the figure on BC is to the sum of the similar and similarly described figures on BA and AC. | V.24
But BC equals the sum of BD and DC, therefore the figure on BC equals the sum of the similar and similarly described figures on BA and AC.
Therefore, in right-angled triangles the figure on the side opposite the right angle equals the sum of the similar and similarly described figures on the sides containing the right angle.
Q.E.D.

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This proposition is a generalization of I.47 where the squares in I.47 are replaced by any similar rectilinear figures. Proclus says that this proposition is Euclid's own, and the proof may be his, but the idea was known to Hippocrates long before Euclid.

Book VI Introduction - Proposition VI.30 - Proposition VI.32.

© 1996
D.E.Joyce
Clark University