I am trying to understand the article by Maryam Mirzakhani about Simple geodesics and Weil-Petersson volumes. In the third section of this article, the following proposition is stated. And I want to know why this proposition is correct. I mention the proposition:

Proposition: Let $P$ be a hyperbolic pair of pants with (non-empty)geodesic boundary components $\beta_1 , \beta_2, \beta_3$ of lenghts $x_1, x_2, x_3$ respectively. Then $P$ contains $5$ complete geodesics disjoint from $\beta_2 , \beta_3$ and orthogonal to $\beta_1$. More precisley, two of these geodesics meet $\beta_1$ respectively at $y_1,y_2$ and spiral to $\beta_3$, the other two meet $\beta_1$ respectively at $z_1,z_2$ and spiral to $\beta_2$. there is also a unique common geodesic perpendicular from $\beta_1$ to itself meeting $\beta_1$ perpendicularly at two points $w_1, w_2$.

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  • $\begingroup$ I don't have time right now to draw pictures, but: To see the unique geodesic from $\beta_1$ to itself, think of the pair of pants as a pair of right-angled hexagon, the saddle geodesic will be apparent. To see the spiraling geodesics, develop the pair of pants into $\mathbb{H}^2$, then you will be able to find a geodesic perpendicular to the developing image of $\beta_1$ and asymptotically approaches the developing image of $\beta_3$. This is in 3.9 of Thurston's notes. $\endgroup$ – Neal Sep 16 at 15:25
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    $\begingroup$ Actually, none of these is a complete geodesic, rather, it is a geodesic ray $\endgroup$ – Moishe Kohan Sep 19 at 14:02

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