Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/217

Rh therefore also EA, AB are in one straight line; [1. 14. join BD.

Then, because the triangle ABC is equal to the trian- gle ADE, [Hypothesis. and that ABD is another triangle, therefore the triangle ABC is to the triangle ABD as the triangle ADE is to the triangle ABD. [V. 7. But the triangle ABC is to the triangle ABD as the base CA is to the base AD, [VI. 1. and the triangle ADE is to the triangle ABD as the base EA is to the base AB ; [VI. 1. therefore CA is to AD as EA is to AB. [V. 11.

Next, let the angle BAC be equal to the angle DAE, and let the sides about the equal angles be reciprocally proportional, namely, CA to AD as EA is to AB: the triangle ABC shall be equal to the triangle ADE.

For, let the same construction be made. Then, because CA is to AD' 'as EA' 'is to AB, [Hypothesis. and that CA is to AD as the triangle ABC is to the triangle ABD, [VI. 1. and that EA is to AB as the triangle ADE is to the triangle ABD, [VI. 1. therefore the triangle ABC is to the triangle ABD as the triangle ADE is to the triangle ABD ; [V. 11. therefore the triangle ABC is equal to the triangle ABD [V. 9.

Wherefore, equal triangles &c.

PROPOSITION 16. THEOREM.

If four straight lines he proportionals, the rectangle contained by the extremes is equal to the rectangle con-tained by the means; and if the rectangle contained by the' extremes he equal to the rectangle contained by the means, the four straight lines are proportionals.