A "Points on a line inequality problem"

[HellloooGuyyys]

I am currently grade 10 and found this math olympiad question in a book a while ago and don't know how to solve it. I try to formulate it myself.
"b) Let A, B, C, D be points in this order on the line g and let P be a point in the plane that is not on g."
Show, if the inequality |AP|+|DP|>|BP|+|CP| holds for all points P that do not lie on g, that |AB|=|CD|"
I appreciate all ideas or solutions

Actually this problem had also an a). The inequality there was |AP|+|DP|=>|BP|+|CP| for als points P on g.
I set the P (1) on the left of A and then (2) on the right of D. And showed that |AB|=>|CD| for (1) and |AB|<=|CD| for (2). It follows that |AB|=|CD|.

But I don't have an idea for b)

 

[Deleted member 4993]

I am currently grade 10 and found this math olympiad question in a book a while ago and don't know how to solve it. I try to formulate it myself.
"b) Let A, B, C, D be points in this order on the line g and let P be a point in the plane that is not on g."
Show, if the inequality |AP|+|DP|>|BP|+|CP| holds for all points P that do not lie on g, that |AB|=|CD|"
I appreciate all ideas or solutions

Actually this problem had also an a). The inequality there was |AP|+|DP|=>|BP|+|CP| for als points P on g.
I set the P (1) on the left of A and then (2) on the right of D. And showed that |AB|=>|CD| for (1) and |AB|<=|CD| for (2). It follows that |AB|=|CD|.

But I don't have an idea for b)

Please show us what you have tried and exactly where you are stuck.

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