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Magic Triangle Puzzle

Date: 07/26/2002 at 17:02:05
From: Pieter
Subject: A flaw in Euclidean geometry


While surfing on the Internet I encountered a site with a triangle 
puzzle on it. The guy that put it on his site claims he has found a 
flaw in Euclidean geometry. You can view the problem here:

   Simeon's Triangle Puzzle 

It looks simple, but I just can't explain it. Can you tell me where 
that white square came from?

Thanks in advance, 

Date: 08/02/2002 at 14:41:07
From: Doctor Nitrogen
Subject: Re: A flaw in Euclidean geometry

Hi, Pieter:

I suspect the reason a blank white square appears in the bottom 
diagram is that in the top diagram, the green right triangle at the 
top and the red right triangle at the bottom are not similar 
triangles. As a result, the area of the entire figure does not remain 
the same when you move the two triangles.

Go back to: 

and examine for yourself. This refers to the top diagram: The acute 
angle at the bottom left of the top green triangle is 

  tan(beta) = 2/5, 

and the acute angle at the bottom left of the larger red 
triangle at the bottom is 

  tan(alpha) = 3/8. 

Put another way, the two smaller sides of the smaller green triangle 
are 2:5, but the ratio of the sides of the larger red one is 3:8. 
Clearly, 2:5 =/= 3:8, and the red and green triangles are not similar.

Why should this matter? Because if the red and green right triangles 
were similar, you could place the green one at the bottom left and the 
red one at the top right, and leave no blank white square when you 
moved the orange figure. But the triangles are not similar, so when 
you place the green one in the lower position and the red one in the 
higher position, you decrease the total area of the larger triangle 
that contains the red and green triangle and the other orange figure. 

In fact, Doctor Ian suggests that for the two triangles to actually 
act as you see on that site, the big triangle that contains both of 
them (as well as the orange figure) would have to have a curved, 
concave-shaped hypotenuse in one case and a curved, convex-shaped 
hypotenuse in the other. 

I did not work out a formal proof for this; nevertheless, that's why I 
suspect the blank white square mysteriously appears in the bottom 

I hope this got you thinking on some intriguing math.

- Doctor Nitrogen, The Math Forum 

Date: 08/02/2002 at 14:43:14
From: Doctor Ian
Subject: Re: A flaw in Euclidean geometry

Hi Pieter,

Dr. Nitrogen's analysis is correct. The key is that the two triangles 
aren't similar. Here's how it looks to me.
In the first triangle, moving from left to right, you move at one
slope (3/8), and then switch to a steeper slope (2/5). This means that 
the 'hypotenuse' is not actually a straight line, but in fact is 
slightly concave. In the second triangle, the situation is reversed: 
you switch from a steeper slope to a gentler one, which makes the 
'hypotenuse' slightly convex.  

In fact, then, neither of the 'triangles' is really a triangle at all, 
but a quadrilateral in which one of the angles is nearly 180 degrees.  
The thickness of the line is used to mask the change in slope; but the 
difference between the convex and concave 'hypotenuses' is the area of 
the white square in the bottom triangle. 

Does that make sense?

- Doctor Ian, The Math Forum 

For more solutions from the Web, see:

   Solution to Triangle Puzzle - Rebecca S. Lindsay 

   Geometric Triangle Puzzle 

   Magic Triangle - Remco Donders 

- Doctor Sarah, The Math Forum 

Date: 08/03/2002 at 03:23:06
From: Pieter
Subject: Thank you (a flaw in Euclidean geometry)

Thank you both for this quick reply! It seems my eyes 
fooled my mind. Thanks for this understandable explanation,

Pieter from the Netherlands

Date: 12/18/2002 at 13:31:32
From: Michael
Subject: Simeon's Triangle Puzzle Answer

Here is a numerical answer, easier to see for some kids:

13 * 5 / 2 = 32.5 square units. That's the apparent size of both 
combined (whole) triangles. But if you calculate the areas of each of 
the sections, again as they appear on the puzzle, they add up to 32 
square units on the top one, and 33 square units on the bottom (the 
one with the white square in it). This confirms that the 'slightly 
concave, slightly convex' analysis is right, and that the hypotenuses 
of both triangles are not straight lines; both are arcs, with a net 
area of one square unit.

Date: 12/18/2002 at 14:41:40
From: Doctor Schwa
Subject: Re: Simeon's Triangle Puzzle Answer

Thanks! Several of the answers we link to are somewhat similar to 
yours, but surprisingly none of them explicitly does the area 

I'd add some of supporting details to your calculation:

Top "triangle" has:
   one 8x3 triangle, area 12
   two "P" shaped pieces, total area 15
   one 5x2 triangle, area 5
   for a total of 32.

Bottom picture has those same four pieces, plus the one square
unit hole, for a total of 33.

Thus neither of them can actually fill the 5x13 triangle exactly.

The other explanations do a good job of showing that the reason they 
don't fit is that the "hypotenuse" is not a straight line. The two 
small triangles have slopes of 3/8 and 2/5, which aren't equal to the 
overall apparent slope of that side, 5/13.

- Doctor Schwa, The Math Forum
Associated Topics:
College Triangles and Other Polygons
High School Triangles and Other Polygons

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