$S$ is set of triangles of unit area. All members of $S$ are uniformly distributed. Let $A$ be the event that a randomly chosen member of $S$ is an isosceles triangle. Prove that the probability of $A$ is zero.
Set up a probability model in which all members of $S$ are uniformly distributed. The choice of the model is up to you. You could view the members of S as members of $R^6$, or $R^3$ or $R^2$, depending on how you choose to describe a triangle. You could even choose a representation in some other set,
Say we have a triangle with sides $a,b$ and angle $x$ between them. Then, since the area is $1$ we have $$a\cdot b\cdot \sin x =2\implies \sin x ={2\over ab}$$
So all we have to chose is $a$ and $b$. Since $0<\sin x\leq 1$ and $a,b>0$ we have $$ b\geq {2\over a}$$
So the sample space is $$\Omega = \{(a,b);\; b\geq {2\over a},\;a>0\}$$
and we are interested in event $$\color{\red}{A =\{(a,b);\; b=a,\;a\geq \sqrt{2}\}}$$