Non-congruent tilings- 20: Rational triangles

The previous episode of this lengthy series is here .

Ref: this mathoverflow discussion .

Broad Question: to tile the plane into rational triangles (all side lengths rational) all mutually non-congruent.

Additional constraints: all triangles should have same area or same perimeter or...

A construction (tiling with mutually non-congruent triangles all of equal area with rationality of side lengths not insisted) presented in detail by Stan Wagon here , it appears, can also be made to yield a tiling of the plane with rational triangles with perimeter unbounded - it appears to give only mutually non-congruent rational triangles, no equality among their areas. A further result obtained by Frettloh (https://arxiv.org/pdf/1603.09132.pdf, more specifically figure 4 therein) indicates that tiling with rational triangles with bounded perimeter also can be got - again with no guarantee on equality of areas.

So, what we could ask for here is for a tiling of the plane into mutually non-congruent rational triangles all with (1)same area or (2) same perimeter or whatever.

An earlier discussion on mathoverflow is here .
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Variant: Rationality of triangles could also be defined as: all angles are irrational fractions of pi. What could one do with them?

Further question: It appears that any polygon with all angles rational can be cut into some finite number of rational angled triangles. Will the question of dividing an n-gon with rational angles into the least number of rational angled triangles have interesting optimization features?

A locus problem

This wasn't received well at Mathoverflow. So here goes:
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Given a line segment AB, the locus of points P such that the angle APB has a constant value is a 'biconvex lens' formed by 2 circular arcs that passes through the points A and B. Special case: if APB is a right angle, then the locus of points is the circle with the diameter AB. Ref: https://mathpages.com/home/kmath173/kmath173.htm#:~:text=Loci%20of%20Equi%2Dangular%20Points&text=Given%20a%20line%20segment%20AB,circle%20with%20the%20diameter%20AB.

Question: Given two line segments AB and CD lying on same plane, what is the locus of points P such that the sum of the angles APB and CPD is a constant? What are the qualitative features of this locus and how does this locus depend upon the relative position, length and orientation of the two segments and the value of the angle sum?

Note: Numerical calculations indicate the following: when AB and CD are kept fixed near to each other or intersecting and the angle sum varied, (1) if the angle sum is small, the locus is a connected curve that lies far away from the two line segments and surrounding them and shows concavities; (2) for larger values of angle sum, the locus lies closer to the two segments and appears convex. And in some cases, for still larger values of the angle sum, the locus probably breaks into two closed curves. Basically, we seem to have a one-parameter family of curves that go from convex to non-convexr>

An analogous question in 3D (with 2 line segments - that could be either skew or coplanar - and an angle sum) could give surfaces as loci of P.
Picture below contours of the angle sum measured with respect to a pair of short and intersecting line segments near the origin