I'm Stuck—In A Curve!

The main thing about a right angle triangle is the 90 degree angle

So for example we take a random point on curve, and place a L shape on it. If we extend the L, we get our right angle triangle. Then we can rotate the L degree by degree and find more triangles.

Since each Jordan curve is all encompassing( goes one whole loop) and does not intersect with itself, half of the rotations of the L always result in right angle triangles, as the other half of the time, the L points to the exterior of the curve.

Hence, simply it is clear that all the points of the curve are vertices of right angle triangles.

ALT: from this Each curve can be made from lines, and each line is part of a right angle triangle as shown. The link shows making the curve from exterior but, is also possible to do from interior. Hence, proven.

We can pick a point A on the curve and draw a straight line into the interior until we hit another point on the curve, which we'll call point B. From point B, we can draw a line perpendicular to the original line in the interior until we hit another point on the curve; call this point C. ABC form a right angle triangle, since there is has a right angle. We are guaranteed to find any three points because the curve is closed and non-intersecting. The starting point A is arbitrary – therefore there are infinite such triangles.