A fallacy is an incorrect result arrived at by apparently correct, though actually specious reasoning. The great Greek geometer Euclid wrote an entire book on geometric fallacies which, unfortunately, has not survived (Gardner 1984, p. ix).
The most common example of a mathematical fallacy is the "proof" that 
as follows. Let 
, then
 |
(1)
|
 |
(2)
|
 |
(3)
|
 |
(4)
|
 |
(5)
|
 |
(6)
|
The incorrect step is (2), in which division by zero (
)
is performed, which is not an allowed algebraic operation. Similarly flawed reasoning
can be used to show that 
,
or any number equals any other number.
Ball and Coxeter (1987) give other such examples in the areas of both arithmetic and geometry.
A geometric example purports to show that every triangle is an isosceles triangle (Ball 1905, pp. 38-39;
Numberphile 2014). Let 
be the intersection of the internal angle
bisector at 
and the perpendicular bisector of 
in a triangle 
with 
. Let 
and 
be the perpendicular feet of 
on the lines 
and 
,
respectively. Auxiliary congruent triangles
give 
and 
. If both feet are
assumed to lie on the side segments, or both on their
extensions beyond 
and 
, adding
or subtracting equal lengths
appears to give 
.
Repeating the argument at the other vertices would then
make the triangle an equilateral
triangle.
The diagram's assumed positions are the error. Suppose 
. The correct construction has 
, so 
lies between 
and 
,
whereas 
lies on the extension of 
beyond 
. Thus 
but 
. One length must be added
and the other subtracted. The auxiliary congruent
triangles are valid, but placing both feet
on the side segments, or both beyond the ends, is
impossible when 
.
