MHB Prove Quadrilateral Divider: Equal Area & Perimeter

[SatyaDas]

Prove that for any quadrilateral there exists at least one straight line that divides the given quadrilateral into 2 equal parts in area and perimeter.

 

[SatyaDas]

I realized I asked in the wrong forum. Don't know how to delete it.

 

[I like Serena]

Moved to Challenge Questions and Puzzles, assuming that was the intended destination.

 

[I like Serena]

Prove that for any quadrilateral there exists at least one straight line that divides the given quadrilateral into 2 equal parts in area and perimeter.

My attempt.

Spoiler

Every line through the centroid of the quadrilateral divides the area into 2 equal halves.
Consider the function that maps the angle of a line through the centroid to the perimeter on one side minus half the total perimeter.
Over a full period, this function is either the zero function, or it has positive maxima with corresponding negative minima.
If it is the zero function, we are done, since any line through the centroid divides the perimeter into 2 equal halves.
So assume that at least 1 positive maximum exists, which means that the perimeter on one side is greater than the perimeter on the other side.
Then there must be a corresponding negative minimum at a distance of half a period from that maximum.
So according to the intermediate value theorem, the function must take the value 0 somewhere between that maximum and minimum.
QED.

 

[SatyaDas]

My attempt.

Spoiler

Every line through the centroid of the quadrilateral divides the area into 2 equal halves.
Consider the function that maps the angle of a line through the centroid to the perimeter on one side minus half the total perimeter.
Over a full period, this function is either the zero function, or it has positive maxima with corresponding negative minima.
If it is the zero function, we are done, since any line through the centroid divides the perimeter into 2 equal halves.
So assume that at least 1 positive maximum exists, which means that the perimeter on one side is greater than the perimeter on the other side.
Then there must be a corresponding negative minimum at a distance of half a period from that maximum.
So according to the intermediate value theorem, the function must take the value 0 somewhere between that maximum and minimum.
QED.

I think this is right solution. Only thing is that I needed to read it multiple times to understand the wordings. :)