Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/220

196 Next, let the rectangle contained by A and C be equal to the square on B: A shall be to B as B is to C,

For, let the same construction be made.

Then, because the rectangle contained by A and C is equal to the square on B, [Hypothesis. and that the square on B is equal to the rectangle contained by B and D, because B is equal to D, [Construction. therefore the rectangle contained by A and C is equal to the rectangle contained by B and D. But if the rectangle contained by the extremes be equal to the rectangle contained by the means, the four straight lines are proportionals; [VI. 16. therefore A is to D as B is to C. But B is equal to D; [Construction. Therefore A is to B as B is to C. [V. 7.

Wherefore, if three straight lines &c.

PROPOSITION 18. PROBLEM.

On a given straight line to describe a rectilineal figure similar and similarly situated to a given rectilineal figure.

Let AB be the given straight line, and CDEF the given rectilineal figure of four sides: it is required to describe on the given straight line AB, a, rectilineal figure, similar and similarly situated to CDEF.

Join DF at the point A, in the straight line AB, make the angle BAC equal to the angle; and at the point B, in the straight line AB, make the angle ABG equal to the angle CDF;[I.23.