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?