Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/214

190 PROPOSITION 12. PROBLEM.

To find a fourth proportional to three given straight lines.

Let A, B, C be the three given straight lines: it is required to find a fourth proportional to A, B, C.

Take two straight lines, DE, DF containing any angle EDF; and in these make DG equal to A, GE equal to B, and DH equal to C; [I. 3.

join GH, and through E draw EF parallel to GH. [I. 31.

HF shall be a fourth propertional to A,B,C

For, because GH is parallel to EF, [Construction.

one of the sides of the triangle DEF, therefore DG is to GE as DH is to HF. [VI. 2.

But DG is equal to A, GE is equal to B, and DH is equal to C; [Construction.

therefore A is to B as C is to HF. [V. 7.

Wherefore to the three given straight lines A, B, C, a fourth proportional HF is found,

PROPOSITION 13. PROBLEM.

To find a mean proportional between two given straight lines.

Let AB, BC be the two given straight lines: it is required to find a mean proportional between them.

Place AB,BC in a straight line, and on AC describe the semicircle ADC; from the point B draw BD at right angles to AC. [I. 11.

BD shall be a mean proportional between AB and BC.