Welcome to our community

Be a part of something great, join today!

Quadrilateral Divider

Satya

New member
Sep 13, 2019
11
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.
 

Satya

New member
Sep 13, 2019
11
I realized I asked in the wrong forum. Don't know how to delete it.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,713
Moved to Challenge Questions and Puzzles, assuming that was the intended destination.
 

Klaas van Aarsen

MHB Seeker
Staff member
Mar 5, 2012
8,713
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.
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.
 

Satya

New member
Sep 13, 2019
11
My attempt.
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. :)