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A Geometric Interpretation Of Equal Sums Of Cubes

A GEOMETRIC INTERPRETATION OF EQUAL SUMS OF CUBES
Abstract:
The geometric interpretation of the Pythagorean Theorem is that the square of diagonal of a rectangle is equal to the sum of squares of two sides. We will discuss some other geometric interpretation of the equivalent forms of equations. The purpose of the work is to find a geometric interpretation of equal sums of cubes in the form
.
Problem: find a geometric interpretation of equal sums of cubes in the form .
Solution:
We know the Pythagorean Theorem:


Let us take a cuboid.

From the triangle ABC:

From the triangle ACD:


Let us cut an angle from this cuboid. We get the corner:

We have .
Let’s put our corner in the zero vector point:

Then A=(0,0,0), D=(a,0,0), C=(0,b,0), B=(0,0,c).

Similarly,







The magnitude of cross product of two vector gives us the area of the parallelogram. So, the area of the triangle BCD is one half from the magnitude of cross product of vectors BD and BC.

Let
p=Area of ∆ABC
r=Area of ∆ACD
q=Area of ∆ABD
s=Area of ∆BCD
Then we see that

But we will try to find something looking like

Let’s take an isosceles triangle ABC, where AB=AC=r, take a point D on BC and take BD=p, CD=q. Without loss of generality take p≤q. Let AD=s.

Let’s draw the altitude AF and the AG=AD. It’s easy to see that GC=p, .

From the triangle ABF

From the triangle ADF





Let’s assume that s=q-p. Then

If we put this result in

we get

Let’s assume that . Then we have the lemma.
Lemma: If ABC is isosceles triangle with AB=BC, and D id the point on BC such that , then

Now we will try to find the geometric interpretation of

Let’s take the right triangle FGH and draw the altitude FJ from the right angle F.




Let K be a point on FH such that . Draw the line though K that is parallel to GH. The point of intersection of this line and FG call L.





Now let’s take the...