# Figure eight geodesic on a pair of pants/Y-piece

Consider a figure-eight geodesic $\delta$( geodesic with exactly one self-intersection point at p ) on a pair of pants Y with three geodesic boundaries $\gamma_i$ and three perpendiculars between them $\alpha_i$. Assume $\delta= \delta_1$ followed by $\delta_2$, where $\delta_i$ are freely homotopic to the geodesic boundaries $\gamma_i$. $\delta_1, \delta_2$ intersect at p.I have two ( basic ) questions :

1. Why does p always lie on $\alpha_3$?
2. Why does $\delta$ intersect $\alpha_i$ orthogonally ?

May be we have to consider the self-isometry of Y which "exchanges" the two right-angled hexagons and has the fixed-point set $\alpha_1 \cup \alpha_2 \cup \alpha_3$ ?

Thank you !

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