Was A1 in player control when he touched the division line and touched the ball?

Simple question Billy never answered.

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Player Control ...
 

Yes (holding or dribbling).

The player never touched the division line and touched the ball at the same time.

Not sure what player control has to do with this situation, other than relating to team control.

This would still be team control even if this somehow became an interrupted dribble.

Isn't this more about ball location rather than player control?

Backcourt 9-9-1: A player shall not be the first to touch the ball after it has been in team control in the frontcourt, if he or a teammate last touched or was touched by the ball in the frontcourt before it went to the backcourt.

If you don't understand what player control has to do with it then I'm done.

What does player control have to do with calling a timeout? Can Team A call a timeout in between dribbles by A1? Or does he have to be touching the ball?

You break the simplest concepts. A player with player control in the front court steps on the division line while still in player control, Billy has nothing.

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Dribbling ...
 

Yes, because A1 has player control (dribbling or holding).

Backcourt 9-9-1: A player shall not be the first to touch the ball after it has been in team control in the frontcourt, if he or a teammate last touched or was touched by the ball in the frontcourt before it went to the backcourt.

Where does 9-9-1 cite player control?

A player that has somehow gotten himself into an interrupted dribble situation (no player control) near the division line still has team control and if said player steps on the division line while in contact with the ball it's a backcourt violation.

Even though rule says that "out-of-bounds violations do not apply on the player involved in the interrupted dribble".

Perfectly Clear ...
 

Not quite.

I'm simply looking for a rule citation that supports a backcourt call, or no backcourt call, if said dribbler is not in contact with the ball when he momentarily steps on the division line.

The present rule language does not make this perfectly clear, as if something had "fallen through the cracks" unnoticed a very long time ago.

Backcourt 9-9-1: A player shall not be the first to touch the ball after it has been in team control in the frontcourt, if he or a teammate last touched or was touched by the ball in the frontcourt before it went to the backcourt.

9-3-1-Note: The dribbler has committed a violation if he/she steps on or outside a boundary, even though he/she is not touching the ball while he/she is out of bounds.

4-9-1: Boundary lines of the court consist of end lines and sidelines.

To support a backcourt call we can either go with ...

NFHS 4-4-3: Ball Location: A ball which is in flight retains the same location as when it was last in contact with a player or the court.

(It's a "stretch" to call a ball between dribbles in "flight".)

... or go with purpose and intent expanding the 9-3-1-Note to include both boundaries and the division line.

Purpose and intent is probably the best bet.

The NBA and WNBA make this situation perfectly clear.

Why can't the NFHS?

If I see this on the court in a few weeks I'm simply calling this a backcourt violation, and players, coaches, fans, and partners will not complain.

But if "shoved into a corner" during an exam study group, I will admit that the situation is not as clear as we may think that it is, and that we may have to rely on the intent and purpose of the 9-3-1-Note.