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April 10th, 2009, 12:07 PM   #1
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need help with geometry proof

this is kind of a hard question to describe, but ill do my best:

let P be any point in the interior of rectangle ABCD (not necessarily the middle). four triangles are formed by joining P to each vertex. demonstrate (prove algebraically) that the area of triangle APD+the area of triangle BPC=the area of triangle APB+the area of triangle PCD. it makes it much easier t o visualize if you draw a simple diagram. thanx for solving this, plz post if you have any questions or need clarification or esp. if you've got the answer!! if you made several diagrams to support your answer, it would be greatly appreciated if you scanned it and sent it to me via email.

your help is appreciated
Antiochus is offline  
April 10th, 2009, 12:29 PM   #2
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Draw the horizontal and vertical lines through P. These lines mark the "height" lines of the four triangles.

The pairs of triangles have bases which are on opposite sides of the rectangle. Assuming AB and CD to be the top and bottom of the rectangle, respectively, the triangles APB and CPD have a combined height that is the same as that of the whole rectangle. Obviously, their bases have lengths that are the same as the rectangle's width. So what is their combined area?

Consider the same questions for the other pair of triangles.

Compare your results.
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April 11th, 2009, 09:57 AM   #3
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Re: need help with geometry proof

Geometry from what I understand is the study of Earth measurement. It is hard to answer yours question algebraically because certain algebra come with it a certain topo. That is, the usage of the word "interior" seems to indicate at least a nature of a topology. For example, at extreme points in considering say R^2, consider point on its boundary, it is uncertain not only in it measurement but also its algebra because it is very much possible that the set of complex number C can literately embeded inside of R. Therefore, because the way it is stated the above and it likes cannot be the case and a geometry must have an equivalence topo represented by an algebra. Locally, geometry is rectangular, this is the use of mathematical quantization. Thus within this quanta, it have bah bah bah; namely, that the area of a triangle is 1/2*b*h.
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