Euclid, Elements
Thomas L. Heath, Sir Thomas Little Heath, Ed.

Therefore the rectangle AE, EB is equal to the rectangle FE, ED.

(Part II. Resolution.)

But the rectangle AE, EB is given, because it is equal to the square on the tangent from E.

Therefore the rectangle FE, ED is given; and, since ED is given, FE is given (in length). [Data, 57.]

But FE is given in position also, so that F is also given. [Data, 27.]

Now FA is the tangent from a given point F to a circle ABC given in position; therefore FA is given in position and magnitude. [Data, 90.]

And F is given; therefore A is given.

But E is also given; therefore the straight line AE is given in position. [Data, 26.]

And the circle ABC is given in position; therefore the point B is also given. [Data, 25.]

But the points D, E are also given; therefore the straight lines DB, BE are also given in position.

Synthesis.

(Part I. Construction.)

Suppose the circle ABC and the points D, E given.

Take a rectangle contained by ED and by a certain straight line EF equal to the square on the tangent to the circle from E.

From F draw FA touching the circle in A; join ABE and then DB, producing DB to meet the circle at C. Join AC.

I say then that AC is parallel to DE.

(Part II. Demonstration.)

Since, by hypothesis, the rectangle FE, ED is equal to the square on the tangent from E, which again is equal to the rectangle AE, EB, the rectangle AE, EB is equal to the rectangle FE, ED.

Therefore A, B, D, F are concyclic, whence the angle FAE is equal to the angle BDE.

But the angle FAE is equal to the angle ACB in the alternate segment; therefore the angle ACB is equal to the angle BDE.

Therefore AC is parallel to DE.

In cases where a διορισμός is necessary, i.e. where a solution is only possible under certain conditions, the analysis will enable those conditions to be ascertained. Sometimes the διορισμός is stated and proved at the end of the analysis, e.g. in Archimedes, On the Sphere and Cylinder, II. 7; sometimes it is stated in that place and the proof postponed till after the end of the synthesis, e.g. in the solution of the problem subsidiary to On the Sphere and Cylinder, II. 4, preserved in Eutocius' commentary on that proposition. The analysis should also enable us to determine the number of solutions of which the problem is susceptible.