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Precalculus Mathematics Homework Help
Diagonals of a Quadrilateral
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[QUOTE="Born, post: 4962875, member: 492924"] 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}## [/QUOTE]
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Diagonals of a Quadrilateral
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