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I actually do not understand where to place this thread. Hope that it is a high school level problem.

There are two triangles ABC and PQR. The vertex A is a middle of the side QR. The vertex P is a middle of the side BC. The line QR is a bisector of the angle BAC. The line BC is a bisector of the angle QPR. Prove that |AC|+|AB|=|PR|+|PQ|.

Using vector algebra methods and huge symbolic calculations in Maple I proved this fact in an additional assumption:

there are no parallel lines among AC, AB, PR, PQ.

I am sure that my method is not adequate and there must be an elementary high-school-level solution but I do not see it.

Please comment.

There are two triangles ABC and PQR. The vertex A is a middle of the side QR. The vertex P is a middle of the side BC. The line QR is a bisector of the angle BAC. The line BC is a bisector of the angle QPR. Prove that |AC|+|AB|=|PR|+|PQ|.

Using vector algebra methods and huge symbolic calculations in Maple I proved this fact in an additional assumption:

there are no parallel lines among AC, AB, PR, PQ.

I am sure that my method is not adequate and there must be an elementary high-school-level solution but I do not see it.

Please comment.

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