Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/74

50 PROPOSITION 47. THEOREM.

In any right-angled triangle, the square which is de-scribed on the side subtending the right angle is equal to the squares described on the sides which contain the right angle.

Let ABC be a right-angled triangle, having the right angle BAC : the square described on the side BC shall be equal to the squares described on the sides BA, AC.

On BC describe the square BDEC, and on BA, AC de- scribe the squares GB,HC; [I.46. through A draw AL parallel to BD or CE ; [I. 31. and join AD, FC.

Then, because the angle BAC is a right angle, [Hypothesis. and that the angle BAG is also a right angle, [Definition 30. the two straight lines AC, AG, on the opposite sides of AB, make with it at the point A the adjacent angles equal to two right angles ; therefore CA is in the same straight line with AG. [I, 14. For the same reason, AB and AH are in the same straight line.

Now the angle DBC is equal to the angle FBA, for each of them is a right angle. [Axiom 11. Add to each the angle ABC. Therefore the whole angle DBA is equal to the whole angle FBC. [Axiom 2. And because the two sides AB, BD are equal to the two sides FB, BC, each to each ; [Definition 30. and the angle DBA is equal to the angle FBC; therefore the triangle ABD is equal to the triangle FBC. [I. 4.