Consequently, the bisection condition to be established in today’s post applies only to convex quadrilaterals — quadrilaterals where none of the interior angles exceed .
Concave quadrilaterals also exist, but we won’t cover those.
Check how many attacks we faced this week! Cause for concern?







As usual, are the side-lengths
. Assume for the time being that none of these sides are equal.
Let the length of diagonal be
. Since
bisects
, both
and
are medians in triangles
and
, respectively. Thus:
The angles and
are equal, so:
We later realized that applying Stewart’s theorem would have made the proof simpler. Same with the converse in example 2 below.







To this end, consider the four triangles ,
,
, and
. For simplicity let’s set
Thus the assumption can then be re-written as
, or
. The angles
and
are supplementary, so:
Similarly, the angles and
are supplementary; so:
The preceding calculation shows that , or
. Thus
and the diagonal
is bisected as desired.



For parallelograms, opposite sides are equal, so we can set and
and obtain
. For squares and rhombuses, we need to take care: pretend that
and
are different, and that
and
are also different; at the same time, use the fact that opposite sides are equal. Doing this gives
.



This is a special case because and
and opposite sides are not equal in general kites. So we’re confronted with the indeterminate form
. However, the longer diagonal in a kite can be as long as possible, and so the ratio
evaluates differently depending on the kite in question. Does this somehow explain why
is said to be undefined?





Let be the point of intersection of
and
. Since
, triangle
is isosceles, and so
is an altitude. Thus
is also an altitude. Since
, this forces
. So we obtain a kite.
Takeaway
Let be a convex cyclic quadrilateral whose diagonals
and
intersect at
. If the side-lengths are
,
,
, and
, then the following statements are equivalent:
bisects
.
Notice the second and third statements.
Task
- (Bisection consequence) Let
be a convex quadrilateral with side-lengths
,
,
, and
. If
bisects
and
(quasi-harmonic), PROVE that
is cyclic.
- (Basic characteristics) Let
be a convex cyclic quadrilateral whose diagonals
and
intersect at
. If
(quasi-harmonic), PROVE that:
-
.