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Subject: "Is Every Parallelogram Rectangle?"     Previous Topic | Next Topic
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nikolinv
Member since Apr-24-10
 Apr-30-11, 12:42 PM (EST)
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"Is Every Parallelogram Rectangle?"
 
   Let a and b denote sides, d1 and d2 denote diagonals of a parallelogram ABCD. Triangle ABC (with sides a, b and d1) and triangle ABD (with sides a, b and d2) have the same area ( same side a and same altitude h).

Let denote H(p,q,r) Heron's formula for a triangle with sides p,q,r.

From H(a,b,d1) = H(a,b,d2) we conclude d1 = d2.

"But this is only possible if ABCD is a rectangle. Is there a paradox, or what?"

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mpdlc
Member since Mar-12-07
 May-03-11, 10:18 PM (EST)
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1. "RE: Is Every Parallelogram Rectangle?"
In response to message #0
 
   Check again the Heron Formula in
https://www.cut-the-knot.org/Curriculum/Geometry/HeronsFormula.shtml
All it comes out that the Area S equals the semi perimeter s times r the incircle radius radius
S = rs
Obviously you have two different triangles with different semi perimeters and different incircle radii even the product of each pair is equal.

mpdlc


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alexbadmin
Charter Member
2803 posts		 May-03-11, 10:22 PM (EST)
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2. "RE: Is Every Parallelogram Rectangle?"
In response to message #1
 
   >Check again the Heron Formula in
>https://www.cut-the-knot.org/Curriculum/Geometry/HeronsFormula.shtml

I have

>All it comes out that the Area S equals the semi perimeter s
>times r the incircle radius radius
>S = rs

>Obviously you have two different triangles with different
>semi perimeters and different incircle radii even the
>product of each pair is equal.

There are more than two triangles. I am only interested in that whose incircle is clearly visible. I do not consider either incircles or inradii of any other triangle.


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alexbadmin
Charter Member
2803 posts		 May-07-11, 02:53 PM (EST)
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3. "RE: Is Every Parallelogram Rectangle?"
In response to message #0
 
   >Let a and b denote sides, d1 and d2 denote diagonals of a
>parallelogram ABCD. Triangle ABC (with sides a, b and d1)
>and triangle ABD (with sides a, b and d2) have the same area
>( same side a and same altitude h).

Also because both triangles are half the parallelogram

>
>Let denote H(p,q,r) Heron's formula for a triangle with
>sides p,q,r.
>
>From H(a,b,d1) = H(a,b,d2) we conclude d1 = d2.
>
>"But this is only possible if ABCD is a rectangle. Is there
>a paradox, or what?"
>
That's actually a good false proof.


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nikolinv
Member since Apr-24-10
 May-09-11, 07:11 PM (EST)
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4. "RE: Is Every Parallelogram Rectangle?"
In response to message #3
 
   This is more algebraic then geometric problem. We need to prove that function H(p,q,x) p,q fixed, is not "one to one", how implicitly claims.

Here is one solution with interesting twist at the end:

Thanks for putting my name in the https://www.cut-the-knot.org/proofs/ParaIsRect.shtml. It is very encouraging for me.


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alexbadmin
Charter Member
2803 posts		 May-09-11, 07:24 PM (EST)
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5. "RE: Is Every Parallelogram Rectangle?"
In response to message #4
 
   >This is more algebraic then geometric problem. We need to
>prove that function H(p,q,x) p,q fixed, is not "one to
>one", how implicitly claims.

Yes, I understtand.

>Thanks for putting my name in the
>https://www.cut-the-knot.org/proofs/ParaIsRect.shtml. It is
>very encouraging for me.

But of course. Thank you.


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