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Math Help - not very hard and not easy, nice geometry

  1. #1
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    not very hard and not easy, nice geometry

    An angle is given (<180 degree) with a vertex X and a point P placed inside this angle. Points A and B lie on different arms of this angle, and XA=XB and the sum PA+PB is the smallest. I ask: when this sum is the smallest? Maybe the angles APX and BPX are equal??

    It is not so easy, that's why i placed it here. Can anybody help me?
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  2. #2
    MHF Contributor red_dog's Avatar
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    I assume that A,B are fixed points such that XA=XB and P is a variable point such that PA+PB is smallest.
    Then P\in[AB] and in this case PA+PB=AB
    If P\not\in[AB], then PA+PB>AB.
    P can be anywhere on the segment AB, so is not necessary that \widehat{APX}=\widehat{BPX}
    Also, the condition XA=XB is not necessary.
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  3. #3
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    Quote Originally Posted by red_dog View Post
    I assume that A,B are fixed points such that XA=XB and P is a variable point such that PA+PB is smallest.
    No, you cannot do that. Firstly we place poin P inside the angle and than we must construct such points A and B. The question is formed properly
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  4. #4
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    Quote Originally Posted by Ununuquantium View Post
    An angle is given (<180 degree) with a vertex X and a point P placed inside this angle. Points A and B lie on different arms of this angle, and XA=XB and the sum PA+PB is the smallest. I ask: when this sum is the smallest? Maybe the angles APX and BPX are equal??

    It is not so easy, that's why i placed it here. Can anybody help me?
    If (PA + PB) is the smallest, then (PA + PB) must be a straight line. The shortest distance between two points, A and B here, is a straight line.

    That means P is anywhere along the line AB.

    That means then that angles APX and BPX are supplementary----their sum is 180 degrees.

    Therefore, (PA + PB) is smallest when angles APX and BPX are supplementary. ---------answer.
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