This is suggested by the question Proof that the sum of the even side and the hypotenuse of a coprime (and positive) Pythagorean triple is a square number and the fact that I really like Euclid's original proof of the Pythagorean theorem (which only uses the fact that the area of a triangle is half the area of a rectangle with the same base and altitude). That proof also proves more, in that it shows how to divide the square on the hypotenuse into two rectangles each of which has the same area as one of the squares on the legs. It also uses no algebra at all.

My question is: Is there a geometrical way to show that a particular right triangle has all integer side?

This assumes, of course, that we are provided with a line segment of length one.

A more basic question is that of deciding geometrically if a line segment has an integer length (given a unit line segment).

The only geometric construction I can think of that might be relevant is that of dividing a segment into a specified number of equal segments (as shown, for example, at https://www.mathsisfun.com/geometry/construct-segment3.html).

An even more basic question is this: If two different constructions each produce a point on a line, are the two points the same?

This can obviously be handled algebraically, since each construction produces a point with ordinate that is a root of a polynomial of degree $2^m$ for some $m$. We then have to decide if the polynomials have a common real root.

I do not know if this question can even be stated in a purely geometrical form.

Well, my question has wandered enough, so I'll stop.

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    $\begingroup$ More fundamentally, what geometric construction allows us to tell whether two given segments are congruent or not? Or given points $A,B,C,$ what construction tells us whether $C$ is on the circle with center $A$ that also passes through $B$? $\endgroup$ – David K Aug 25 '17 at 2:13

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