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A quadrilateral with one diagonal that bisects opposite angles and another diagonal that does not bisect opposite angles may or may not be a kite.

We know that a kite is a quadrilateral if it has two pair of equal and adjacent sides, of which the diagonals intersects at right angles, but how could we conclusively prove the validity of this statement?

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This may depend on whether a (non-convex) symmetric dart counts as a kite. The words "is" in the question and "would" in the statement look a bit odd. – Henry Feb 2 '12 at 17:37
The statement is logically of the form "$A$ or not $A$" so it is a tautology, and trivially valid. What do you really mean to ask? – Gerry Myerson Feb 2 '12 at 23:20
@GerryMyerson - This proposition is not a tautology. It says that a quadrilateral with this property may be a kite, and that a quadrilateral with this property may be not a kite. If all quadrilaterals with this property are kites, or if all quadrilaterals with this property are non-kites, then this proposition is false. To prove the validity of this proposition, one would have to exhibit a kite with this property, and also a non-kite with this property. – user22805 Feb 4 '12 at 9:18
@DavidWallace, we have different ideas about mathematical English. I interpret "a square may or may not have five sides" as "(a square may have five sides) or (a square may not have five sides)" and thus true, even though there isn't a square with five sides. – Gerry Myerson Feb 4 '12 at 11:34
@GerryMyerson - then maybe the OP needs to clarify which interpretation he/she has in mind. – user22805 Feb 4 '12 at 21:39

Since I can't draw, I will use coordinates, and you can do the drawing.

The quadrilateral clearly can be a kite. For completeness, we show this. Let the vertices of our quadrilateral, in counterclockwise order, be $A(1,0)$, $B(0,2)$, $C(-1,0)$, and $D(0,-1)$. This is a kite, and the diagonal $BD$ bisects a pair of opposite angles, and the diagonal $AC$ doesn't.

Now let's produce a suitable non-kite $ABCD$. What is a kite? Does it have to be convex? If it does, here is an example of a non-kite with the desired properties. Let the vertices be $A(1,0)$, $B(0,2)$, $C(-1,0)$, and $D(0,1)$. Note that this is non-convex, the part $CDA$ sticks in, not out.

If you are limiting attention to convex quadrilaterals, then, as was pointed out by Henry, there are no non-kites which have the property that one diagonal bisects a pair of opposite angles. For the diagonal that bisects a pair of opposite angles divided the quadrilateral into two triangle, which can be shown to be congruent (they have a common side, and all corresponding angles match). Thus sides match in pairs. If the quadrilateral is convex, this forces it to be a kite.

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