How do we know the lengths of the sides of a triangle?

[Icebreaker]

Besides empirical measuring and without Pythagoras' knowledge of the theorem, how do we know the lengths of the sides of a triangle? Is this dealt with in "The Elements"?

That is, if we define the length of one side, with three angles, how do we know what's the measure of the other two sides using the most elementary methods?

 

[James R]

As far as I am aware, there are no more elementary methods for solving that kind of problem than by using trigonometry.

 

[Crosson]

Suppose we are given a length L for the hypotenuse of this right triangle, and an angle a. Then the other two sides of the triangle are given by:

$$L_1 = L (a- \frac{a^3}{3!} + \frac{a^5}{5!}...)$$

$$L_2 = L (1+ \frac{a^2}{2!} - \frac{a^4}{4!}...)$$

This can be deduced from a knowledge of polynomials, and viewing the sine of the angle as a function. It would be interesting to derive these formulae by a purely geometric argument.

 

[what]

I believe if you take each side of a right triangle and draw a square, three sqaures in total each with a side L1, L2, L3, you should be able to see the relationship. I don't know if that was what you were asking but here's a link to show what i meant http://jwilson.coe.uga.edu/emt669/Student.Folders/Morris.Stephanie/EMT.669/Essay.1/Pythagorean.html .