[pringk02]

> do we know that both diagonals are diameters

This must be true, because the diagonals are both straight lines that go through the centre and are bound by the edges, so it follows they must be equal to the diameter of the circle by definition.

> Do we know that a parallelogram with equal diagonals is a rectangle?

As another commenter points out, this is a theorem you can reach for, but proving it by itself is a bit more of a task.

[HarHarVeryFunny]

> This must be true, because the diagonals are both straight lines that go through the centre

How do we know the 2nd diagonal goes through the center ? Is it because of the construction by rotation ?

[surajms]

Yes. So, here when we rotate the triangle, we are essentially rotating each of the endpoints. For each endpoint, we rotate it by 180 degrees around the line segment joining the endpoint and the center. This by definition will result in a new position for each endpoint that creates a chord (as the two endpoints lie on the circle) and passes through the center (we rotated around it). A chord that passes through the center is by definition a diameter.

[HarHarVeryFunny]

Thanks. Thinking about it, another way of looking at it is to construct the rotated triangle by drawing a line from each point through the center to where it intersects with the circle on the other side. This is obviously a 180' rotation, but by construction we explicitly know the diagonal goes through the center.

[lupire]

It maybe more clear if you visualize the reflection as a pair of perpendicular reflections, first across the diameter (which is also across the center) and then internally reflecting the diameter (which is again also across the center.)

Two reflections with a common fixed point make a rotation around that fixed point (angle of reflection is double the angle between the reflection axes.). Two perpendicular reflections make a 180 degree rotation around the intersection of the axes of rotation.