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### Magic Triangle Puzzle

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

Hello,

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
http://www.simeonmagic.com/triangle/triangle1.htm

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

Pieter
```

```
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:

http://www.simeonmagic.com/triangle/triangle1.htm

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
diagram.

I hope this got you thinking on some intriguing math.

- Doctor Nitrogen, The Math Forum
http://mathforum.org/dr.math/
```

```
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
http://mathforum.org/dr.math/

For more solutions from the Web, see:

Solution to Triangle Puzzle - Rebecca S. Lindsay
http://www.loisterms.com/trisolution.htm

Geometric Triangle Puzzle

Magic Triangle - Remco Donders
http://gene.wins.uva.nl/~donders/magic.htm

- Doctor Sarah, The Math Forum
http://mathforum.org/dr.math/
```

```
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

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
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
http://mathforum.org/dr.math/
```
Associated Topics:
College Triangles and Other Polygons
High School Triangles and Other Polygons

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