Use similar triangles.

This is the converse of the previous statement in example 1. One proof of it uses similar triangles and the law of sines.

By example 1 we have that . If bisects , then . And so . Similarly for the converse.

By example 1 we have that . If bisects , then . And so . Similarly for the converse.

This follows from example 3 and example 4 above, since and yield and .

## Brief relief

*hackers*and their bots: here, here, and here. Good thing is that we have the upper hand, as at the time of this writing.

## Takeaway

Let be a convex cyclic quadrilateral. Then the following statements are *equivalent*:

- is a rectangle
- diagonals and bisect each other.

## Task

- (Easy verification) Let be a convex cyclic quadrilateral with vertices at , , , . Verify that:
- diagonal bisects diagonal
- .