PROPOSITION 31. THEOREM.
In any right-angled triangle, any rectilineal figure described on the side subtending the right angle is equal to the similar and similarly described figures on the sides containing the right angle.
Let ABC be a right-angled triangle, having the right angle BAC: the rectilineal figure described on BC shall be equal to the similar and similarly described figures on BA and CA.
Draw the perpendicular AD. [I. 12.
Then, because in the rightangled triangle ABC, AD is drawn from the right angle at A, perpendicular to the base BC, the triangles ABD, CAD are similar to the whole triangle CBA,and to one another. [VI. 8.
And because the triangle CBA is similar to the triangle ABD,
therefore CB is to BA as BA is to BD. [VI. Def. 1.
And when three straight lines are proportionals, as the first is to the third so is the figure described on the first to the similar and similarly described figure on the second; [VI. 20, Corollary 2.
therefore as CB is to BD so is the figure described on CB to the similar and similarly described figure on BA;
and inversely, as BD is to BC so is the figure described on BA to that described on CB. [V. B.
In the same manner, as CD is to CB so is the figure described on CA to the similar figure described on CB.
Therefore as BD and CD together are to CB so are the figures described on BA and CA together to the figure described on CB. [V. 24.
Cut BD and CD together are equal to CB;
therefore the figure described on BC is equal to the similar and similarly described figures on BA and CA. [V. A.
Wherefore, in any right-angled triangle &c. q.e.d.
