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
.
Brief relief
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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
.
- diagonal