football

That's my point; I'm not assuming it is a right triangle, but when the construction is drawn I (even if not you) can see that it must be one. I've even given you a couple of interim steps to seeing why this is so, but you don't seem to grasp it even then.

Maybe you can't see it, but I can.

I started with a triangle in a semi-circle as requested by OP and did not assume it was a right-angled triangle. However when completing the construction as I described, it is clear (to me at least) that it can only have been a right angle to start with. I filled out a couple of extra steps to help you complete the visualisation, but it seems you need the formal algebra. Maybe OP still needs the proof, but to me it is so obvious as not to need formal proof.

It does not assume that. Maybe it doesn't work for you, but I can see from this thought experiment, without needing the standard proof, that it can only be a rectangle.

If you need some interim steps between looking at the image and concluding that it is a rectangle (getting nearer to the proof that OP specifically said he didn't want) you can imagine trying to deform the quadrilateral to vary the angles while still maintaining the vertices on the circle, you can see that this is impossible. The quadrilateral is fixed. Looking at the symmetry again shows that all the angles are right angles.

Since the starting triangle was a random angle in a semi-circle, the visualisation applies to all such triangles.

Can you not see this without needing an algebraic or verbal proof?

Religions are belief systems that either reproduce themselves or die out. Like genes there will be a process that ensures or fails to ensure survival.

1. Judaism reproduces itself primarily by attachment to the Jewish race, hence the confusion as to whether the term might refer to a religion or a race. This confusion does not apply to the other two.

2. Christianity primarily reproduces and multiplies itself by conversion (preaching) and also by passing down the generations.

3. Islam reproduces itself by forcible conquest and then passing down the generations.

You might like to visualise this way:

1. Draw your triangle.
2. Draw the triangle you get from rotating the fist triangle 180 degrees about the centre of the circle.
3. You can see, without doing the work of proving it, that what you have constructed can only be a rectangle and all four corners will be of 90 degrees.