How does MTW derive $s_{AB}^2 =−\tau_{AB}^2 = −\tau_{AQ} \tau_{AP}$ from the radar construction?

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$How does MTW derive $s_{AB}^2 =−\tau_{AB}^2 = −\tau_{AQ} \tau_{AP}$ from the radar construction? 图片 1$

$How does MTW derive $s_{AB}^2 =−\tau_{AB}^2 = −\tau_{AQ} \tau_{AP}$ from the radar construction? 图片 2$

$How does MTW derive $s_{AB}^2 =−\tau_{AB}^2 = −\tau_{AQ} \tau_{AP}$ from the radar construction? 图片 3$

I am reading Gravitation by Misner, Thorne, and Wheeler (MTW), Chapter II ("Local Lorentz Geometry"), and I am confused about the derivation of the equation
$
s_{AB}^2 =−\tau_{AB}^2 = −\tau_{AQ} \tau_{AP}.
$
The setup is shown in the figure below:

A free particle follows the worldline AX. At event P, it emits a light signal toward event B, and the reflected light returns to the worldline at event Q.

I think I understand that:

The particle measures only the proper times $\tau_{AP}$ and $\tau_{AQ}$

The light is emitted at P and received back at Q,
the construction is related to Einstein's radar method.

However, I do not understand the following points:

Why is the interval being computed between A and B rather than between P and B, since the light signal is actually emitted from P?

Where does the intermediate quantity $\tau_{AB}^2$ come from? Since B is not on the particle's worldline, what exactly does $\tau_{AB}$ mean here?

Is MTW implicitly choosing the particle's rest frame or using Einstein synchronization/radar coordinates before writing this equation? If so, where does that assumption enter the derivation?

I would appreciate a step-by-step explanation of the logic rather than just the final formula.

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How does MTW derive $s_{AB}^2 ​ =−\tau_{AB}^2 ​ = −\tau_{AQ} ​\tau_{AP}$ ​ from the radar construction?

How does MTW derive $s_{AB}^2 =−\tau_{AB}^2 = −\tau_{AQ} \tau_{AP}$ from the radar construction?