Let $2\le a\le b\le c\in \mathbb{N}$ with $\mu =1/a+1/b+1/c<1$ and let $T=T_{a,b,c}=\langle x,y,z:x^a=y^b=z^c=xyz=1\rangle$ be the corresponding hyperbolic triangle group. Many papers have been dedicated to the following question: what are the finite (simple) groups which appear as quotients of $T$? (Classically, for $(a,b,c)=(2,3,7)$ and more recently also for general $(a,b,c)$.) These papers have used either explicit constructive methods or probabilistic ones. The goal of this paper is to present a new approach based on the theory of representation varieties (via deformation theory). As a corollary we essentially prove...