In non-euclidean geometry, can hypotenus of rightangle triangle be shorter than sides? Disclaimer. This is only a recreational question in geometry...

In euclidean geometry, the following picture is definitely inconsistent!

Of course, the issue is that, the picture suggests that the diagonal is at most as long as the sides, i.e $\mathbf{\sqrt{2} \le 1}$ (a contradiction!).

Question (loose). Is there a (necessarily non-euclidean) geometry in which the picture would be consistent ?

Noneuclidean geometry
To make sense of the above question, one needs to first extend certain euclidean geometry. Viz


*

*A straight line is a geodesic curve.

*Two straight lines $\mathcal L_1$ and $\mathcal L_2$ are perpendicular at a point $A$ if their tangents at $A$ are perpendicular

*A square $ABCD$ of side length $L$ is a quadruple of four lines (i.e geodesic curves!) $AB,BC,CD,DA$ such that


*

*Perpendicular sides. $AB \perp BC$ at $B$, $BC \perp CD$ at $C$, and $CD \perp DA$ at $D$, $DA \perp AB$ at $A$.

*Equal side lengths. $\ell(AB) = \ell(BC) = \ell(CD) = \ell(DA) = L$.



The diagonals of a square are the straight lines (i.e geodesics) $AC$ and $BD$.
Question (refined). Is there geometry (i.e smooth riemannian manifold) in which there exists a square of side length $L$ both of whose diagonals have (equal) length which is $L$ or less ?
 A: A hypotenuse by definition is the longest side in a right triangle (which is opposite the right angle). In flat geometry, such triangles can have only one right angle, hence only one hypotenuse.
In geometries where the sum of angles in a triangle is less than $180°,$ there will also be one hypotenuse. However, in geometries where the sum of the angles exceeds this value, then there may well be more than one hypotenuse. Also, since in this kind of geometry, a triangle may be simultaneously right-angled and obtuse angled, then it is possible for a side to be longer than each of the sides opposite the right angle. Thus this answers your question in the positive. However such questions are already moot here since the Pythagorean theorem doesn't hold, so what's the point of defining hypotenuses same as they are defined in flat geometry, where they are singled out there because of their importance and the many relationships they reveal?

OK, I've read the edited question, and the germ of the answer is contained in the above answer -- I'll highlight this. Your question is equivalent to asking if there exist geometries where an isosceles triangle with only one right angle (where the angle is contained between the equal legs) has the base not more than the equal legs. Again, the answer is that that happens in some geometry where the angle sum in triangles is more than $180°,$ for then we would have a case where the base angles are equal to or more than $90°$ each, which would make the legs of the triangle be equal to or even exceed that of the side opposite the unique right angle.
