CIIM 2011 Problem 6

Let $\Gamma$ be the branch $x> 0$ of the hyperbola $x^2 - y^2 = 1.$ Let $P_0, P_1,..., P_n$ different points of $\Gamma$ with $P_0 = (1, 0)$ and $P_1 = (13/12, 5/12)$ . Let $t_i$ be the tangent line to $\Gamma$ at $P_i$ . Suppose that for all $i \geq 0$ the area of the region bounded by $t_i, t_{i +1}$ and $\Gamma$ is a constant independent of $i$ . Find the coordinates of the points $P_i$ .