If QB drops the ball on a fake handoff then picks it up and throws a pass...

Because its not illegal. You can fumble the ball and pick it up and throw the ball so long as you're behind the line of scrimmage

Why would it be an illegal forward pass? Not past the LOS or a second one.

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Could an offensive lineman do the same?

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No it would be illegal touching

this can't be serious.

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Could an offensive lineman do the same?

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In order to throw a forward pass, the person throwing the pass must meet certain criteria.

1. They must be behind the line of scrimmage

2. There cannot have been a forward pass already thrown on the down

3. The person throwing the pass must be an eligible receiver (the QB is technically an eligible receiver). So an offensive lineman cannot throw a forward pass.