Perpendicular bisectors of the sides in a quadrangle

[Stylewriter]

1. What reqiurements does a quadrangle need that all the perpendicular bisectors of the sides crosses in one point??

2. And how can you proof it?

To 1. - I think each angle have to be 90° but I can't proof it

 

[BobG]

Off the top of my head, I couldn't tell you the requirements, but all angles equal to 90 definitely isn't one. The perpendicular bisectors of a trapezoid will meet in one place if both angles on the base are equal and both angles on the top are equal.

Edit: Once I think about it, the requirements are pretty obvious (in fact, I think I did know this off the top of my head at one point). Start from the intersection, drawing the perpendicular bisectors for two of the sides. However long your sides have to extend in one direction in order to form an intersection, they have to extend in the opposite direction as well. Then start working on the last two sides.

The relationship between the angles is pretty straight forward - A rectangle with 4 90 degree angles meet them, but it's not the only quadrangle to meet them. (That should limit the possible relationships down to a pretty small number).

 

[thepatientmental]

To 1. The requirement for the perpendicular bisectors of the sides in a quadrangle to intersect at one point is that the quadrangle must be a parallelogram. This means that opposite sides are parallel and equal in length, and opposite angles are also equal. This can be proven using the properties of a parallelogram, such as the opposite sides being equal in length and the diagonals bisecting each other.

To 2. To prove that the perpendicular bisectors of the sides in a quadrangle intersect at one point, we can use the properties of a parallelogram.

First, we can show that the opposite sides of a parallelogram are equal in length. This can be proven by the fact that a parallelogram has two pairs of parallel sides and opposite sides are congruent.

Next, we can show that the diagonals of a parallelogram bisect each other. This can be proven by the fact that the opposite sides of a parallelogram are congruent, so the two triangles formed by the diagonals are congruent by SAS (side-angle-side) congruence. Therefore, the diagonals bisect each other at their intersection point.

Since the perpendicular bisectors of the sides in a quadrangle are also the diagonals, we can conclude that they will intersect at one point in a parallelogram. This is because the properties of a parallelogram show that the opposite sides are equal in length and the diagonals bisect each other, which means that the perpendicular bisectors of the sides will also intersect at one point.