# Impossible triangles, in what sense do they exist?

In what sense do impossible triangles and their properties exist, if they do at all?

There is a rule for right triangles which states that the altitude of the hypotenuse can't be greater than half of the hypotenuse.

(Don't get confused with the notation of these pictures, the same letters in different pictures mean different things) In the first of the two pictures, the altitude of the hypotenuse is the line AC. From the second picture you can see that if you move the point B on the circle, the altitude of the hypotenuse is at it's greatest when the point B is directly above the point O (when OB is perpendicular to AC). At that point the altitude of the hypotenuse if exactly the radius of the circle, which is half of the diameter of the circle, and thus half of the hypotenuse.

So you can't have a triangle with the altitude of the hypotenuse exceeding half of the length of the hypotenuse. So if the length of the hypotenuse is 4, the altitude of the hypotenuse can be at most 2. But what if we calculate the area for a triangle with hypotenuse of length 4 and the altitude of the hypotenuse of length 3? The area of a triangle is its base times its height divided by two, so our triangle's area would then be (4x3)/2 = 6

But what does it mean? No such triangle could exist, but if it would, its area would be 6? What does it mean to calculate a property for something that doesn't exist?

• There a paragraph in Kant that deal exactly with 'impossible triangles' which is a corollory of his notion of space, and what he means by an analytic proposition. Feb 29 '16 at 12:53
• While Jo Wehler is correct, I believe there may be other "geometries" in which such objects can be said to exist. I'm not sure, but it seems that just as geometry can be captured in algebra, it may be that a correct algebraic statement such as 4x3/2 = 6 could be a basis for some different sort of redefined "triangular area." The "impossible" is a defined contradiction within some analytically coherent system, and may be "possible" in some other system, though such relativism is not very useful. Feb 29 '16 at 15:32
• There's such a thing as inconsistent geometry; and here "it is an investigation of inconsistency, in a suprising area...geometry, that paragon of clear and distinct ideas [becomes] a site of contradiction; the book is uncompromising and technically demanding"; which, on reflection, is unsurprising for a theme that sails upwind against tradition. Mar 1 '16 at 4:20