Prove Quadrilateral Divider: Equal Area & Perimeter

  • Context: MHB 
  • Thread starter Thread starter SatyaDas
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Discussion Overview

The discussion revolves around the proposition that for any quadrilateral, there exists at least one straight line that can divide the quadrilateral into two equal parts in both area and perimeter. The scope includes mathematical reasoning and problem-solving related to geometry.

Discussion Character

  • Mathematical reasoning, Debate/contested

Main Points Raised

  • One participant proposes the existence of a straight line that can achieve equal area and perimeter division for any quadrilateral.
  • Another participant expresses uncertainty about the initial question's appropriateness for the forum.
  • A third participant moves the thread to a more suitable category, indicating a potential misunderstanding of the original context.
  • Subsequent posts include attempts to prove the proposition, but details of these attempts are not fully elaborated.
  • One participant mentions needing to read the solution multiple times to grasp the wording, suggesting complexity in the explanation.

Areas of Agreement / Disagreement

The discussion does not reach a consensus, as participants have not fully engaged with the proof attempts or provided definitive responses to the initial proposition.

Contextual Notes

Some assumptions about the properties of quadrilaterals and the definitions of area and perimeter may be implicit but are not explicitly stated. The clarity of the proof attempts remains unresolved.

SatyaDas
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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.
 
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I realized I asked in the wrong forum. Don't know how to delete it.
 
Moved to Challenge Questions and Puzzles, assuming that was the intended destination.
 
Satya said:
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.
 
Klaas van Aarsen said:
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. :)
 

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