Let's make one last summary of all below.
Since I learned on my own to address problems in Euclidean geometry from a purely synthetic approach, I've been fascinated by it. If only I had learned this on my teenage years...
Yet this time the passion for a visual approach seems to have mislead me down a windy rabbit hole just come out at what seems trivial had I chosen a more analytical approach of counting degrees of freedom.
We need 6 numbers to specify the coordinates of three points in the plane. If it's a right triangle, only five coordinates are independent.
But we don't have to stick to cartesian coordinates.
If we fix one point on the origin, we need four number for the two remaining points plus two more for defining the translation that sat this triangle on the origin.
We can repeat that thought process and further constrain a second point to be on the vertical axis. We can bring it there by a rotation, after using a translation to fix the first point. This leaves then three coordinates for one point "and a half", plus a number for specifying the rotation we had to apply, and two more numbers for the translation. For a right triangle, the third point is constrained as well, in this case, to be on the horizontal axis. Again it amounts to 6 and 5 dof, respectively.
If we fix the length of the "vertical" side to 1, we reduce the previous count of coordinates by one, which we use to specify a dilation. Fir right triangles then we have one horizontal coordinate, one scaling factor, one rotation, and the two numbers for a translation.
And this is nothing more what all the rambling below amounts to.
So, yeah, we have reduced a right triangle to a single point on a line; the whole 2d figure "emerges" from that single point. But the number of degrees of freedom have not changed, we simply hid them in the form of transformations.
I think one take home lesson is to always explore a problem from multiple perspectives, e.g, synthetic & analytic , very soon at the start. Some paths offer much less resistance for making solid steps forward.
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