Diagonals of a Quadrilateral

  • #1
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Homework Statement



Problem 55 from Kiselevś Geometry - Book I. Planimetry: "Prove that each diagonal of a quadrilateral either lies entirely in its interior, or entirely in its exterior. Give an example of a pentagon for which this is false."

Homework Equations




The Attempt at a Solution



The pentagon part is pretty easy. I'm having trouble with the proof. A proof by contradiction seems to be the easiest way to solve this problem but I'd prefer a proof that also explains why this should be true.

I've tried using straight line properties (i.e. a straight line can be formed though any two points and it is unique) but I haven't gottten anywhere.

Thanks in advanced for any help!
 
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Answers and Replies

  • #2
Simon Bridge
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Can you see why the pentagon can break the rule?
If a diagonal breaks the rule, then it must cross one of the sides - that help?
You can also look at the classes of quadrilateral and see how diagonals are formed in each case.
 
  • #3
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Maybe supplementary angles of a transversal are....
 
  • #4
lurflurf
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the diagonal divides the plane in two and contains exactly two of the four points there are two cases both points lie on the same side or each lies on one side
 
  • #5
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Simon, MidgetDwarf, and lurflurf, I think you'll like what I've come up with. I'm sorry to not be able to show some pictures but I believe the written proof will suffice. Hope it's clear enough. Thank you for your help.

The three properties of straight lines in the proof are the following: (1) A straight line can be created from any two points, (2) this line is unique, and (3) if two straight lines coincide at least at two points, all their points coincide (making them the same line).

##\mathrm{Proof:}##

A quadrilateral has four vertices, each vertex point must connect to two others in order to form the sides of the quadrilateral. Labeling these four points A, B, C, and D and forming the following sides AB, BC, CD, and DA we create the quadrilateral ABCD. The diagonals of said quadrilateral will consequently be AC and BD.

If a diagonal were to not lie completely inside or outside the quadrilateral then it (the diagonal) must cross one of the sides of the quadrilateral (either to enter or to exit the figure).

The diagonal AC cannot cross the side AB, DA, BC, or CD because this would imply that the diagonal AD equals the respective side it crosses by property (3) (since AC would coincide with the point of the side it crosses and the point A or C). The same applies to BD and the side AB, DA, BC, or CD.

This implies that the diagonals of a quadrilateral cannot cross its sides.

Therefore the diagonals of a quadrilateral must either lie entirely inside or entirely outside. ##\mathrm{QED}##
 
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  • #6
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##\mathrm{Follow\ up:}##

Concerning the pentagon: labeling the five vertex points A, B, C, D, E; and forming the sides AB, BC, CD, DE, and EA. A diagonal is made from point A to point D crossing the side BC. This is possible since the diagonal would only share one point with the side BC- (This is unlike the quadrilateral in which every diagonal would share two points of a side if said diagonal crossed said side)

Therefore a diagonal which lies partially outside and partially inside the figure is possible. ##\mathrm{QED}##
 
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