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.

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(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?

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    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
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    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

You cannot determine a property of something which does not exist. Properties need a subject whose properties they are.

But what if we calculate the area for a triangle with hypotenuse of length 4 and the altitude of the hypotenuse of length 3?

You cannot do this because such triangle does not exist. Your sentence is contradictory because already the concept of a triangle with hypotenuse of length 4 and the altitude of the hypotenuse of length 3 is a contradictory concept.

When forming concepts the first rule is to assure that they are not contradictory. Otherwise the concept is without meaning.


Specifically, based on the question as you have it written I would say that you are pulling a mental bait and switch, you begin the problem with a proposition that is true of right triangles, and you then use values that will not satisfy the proposition for right triangles using a formula which is true for all triangles. Your conclusion that the values generate an impossible triangle is not correct, what you could have proven with those dimension values is that you could not form a right triangle with those values. So there is no issue with the values, or the resultant area of a triangle. If you created an area equation for possible right triangles, then you would find that these values are not in the solution set.

Having said that, my next question would be, is it possible to generate a right triangle with those dimension values? I am pretty sure that I just said it is not, but does that original proposition hold true if the right triangle is on a 3 dimensional surface like the curve of a bubble sitting on a flat surface? If one were able to alter the surface in the way that a bubble or droplet of water sits on a surface, then the amount of curvature of each line could be altered while retaining the right angle by managing how curved the surfaces are ranging from a sphere back to the 2 dimensional surface. I haven't had time to work this out, or look for solutions that have been completed, but thinking in 3 dimensions may lead to a solution.

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