Page:The Elements of Euclid for the Use of Schools and Colleges - 1872.djvu/330

306 the excess of its radius over the radius of the circle with centre A will touch externally the three given circles.

In a similar way we may describe a circle touching internally the three given circles, or touching one of them externally and the two others internally, or touching one of them internally and the two others externally.

17. In a given indefinite straight line it is required to find a point such that the sum of its distances from two given points on the same side of the straight line shall be the least possible.

Let A and B be the two given points. From A draw a perpendicular to the given straight line meeting it at C and produce AC to D so that CD may be equal to AC Join DB meeting the given straight line at E. Then shall be the required point.

For, let F be any other point in the given straight line. Then, because AG is equal to DC, and EC is common to the two triangles ACE, DCE; and that the right angle ACE is equal to the right angle DCE; therefore AE is equal to DE. Similarly, AF is equal to DF. And the sum of DF and FB is greater than BD (I. 20): therefore the sum of AF and FB is greater than BD; that is, the sum of AF and FB is greater than the sum of DE and EB; therefore the sum of AF and FB is greater than the sum of AE and EB.