The presentation shows a new demonstration of Pythagorean theorem that does not use the Low of Cosine

1. Generalized Pythagorean Theorem
Danut Dragoi, PhD
California
04/15/2017

2. Proofs of Pythagorean Theorem
There are various proofs of Pythagorean theorem at these links
https://en.wikipedia.org/wiki/Pythagorean_theorem
http://www.cut-the-knot.org/pythagoras/index.shtml
The author of the article “The Law of Cosines (Cosine Rule) “ post-it on this site:
http://www.cut-the-knot.org/pythagoras/cosine.shtml
(Alexander Bogomolny) mentioned in his article that “ I'll be extremely curious
to learn of any proof of the Cosine Rule completely independent of the
Pythagorean Theorem.”
The expression to be demonstrated can be seen at this site:
http://www.nabla.hr/GE-AppTrigonomA4.htm
This theorem is also known as cosine theorem, see the link above. For right
triangles the usual Pythagorean theorem has a sum of two squared terms.
For the other triangles, the expression of the square of any side of the
triangle contains an additional term that we can call it for generalized
Pythagorean theorem, which will be proved in this presentation using the
"power-of-a-point theorem"
https://www.google.com/#q=power+of+a+point+outside+a+circle.
.

3. 3
For any triangle ABC, draw the circle with the center on the point B and radius AB=c see
the Figure below. Extend the line CA to cross the circle second time on the point A'.
Project the point B on CA' line and obtain the point B'. Using the traditional notation
BC=a, AB=c, AC=b, and angle BAC=α we can demonstrate the general expression for
one side length for any type of triangle as a function of the other two sides and the angle
between them.
New proof of generalized
Pythagorean Theorem

4. 4
Applying the power of point C from the circle centered in point B
we get the expression:
CP*CP'=CA*CA'
(a-c)*(a+c)=b*(b+2c*cos(180-α))----------(1)

5. 5
If the triangle is oblique, angle α<90, the power of point C has an
equivalent expression as before, the only difference from the first
derivation is that the point C is interior to the circle.
CP*CP'=CA*CA'
(c-a)*(a+c)=b*(2c*cosα−b) ---------(1')

6. 6
Opening the parentheses we get the same expression from both
cases of the generalized Pythagorean theorem:
(c-a)*(a+c)=b*(2c*cosα−b) ---------(1)
a2
=b2
+c2
-2bc*cosα------(2)

7. 7
Conclusion
-The method “power of a point to a circle”
provides a good proof for the generalized
Pythagorean theorem
-Power of a point to a circle method is
appropriate to prove Pythagorean
theorem for all types of triangles
(generalized theorem).