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# Bisecting a diagonal in a cyclic quad

Let be a cyclic quadrilateral with side-lengths , , , and . Let and be the diagonals. Then bisects if and only if , and bisects if and only if .

Let be a convex quadrilateral whose diagonals intersect at . If is cyclic, PROVE that and .

Use similar triangles.

Let be a convex quadrilateral whose diagonals intersect at . If and , PROVE that is cyclic.

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

Let be a convex cyclic quadrilateral whose diagonals intersect at . PROVE that diagonal bisects diagonal if, and only if, .

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

Let be a convex cyclic quadrilateral whose diagonals intersect at . PROVE that diagonal bisects diagonal if, and only if, .

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

Among convex cyclic quadrilaterals, PROVE that only rectangles have the two diagonals bisecting each other.

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

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## Takeaway

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

1. is a rectangle
2. diagonals and bisect each other.

## Task

• (Easy verification) Let be a convex cyclic quadrilateral with vertices at , , , . Verify that:
1. diagonal bisects diagonal
2. .