Pentagon with two right angles (aka Van Aubel's Theorem)Given that ABCDE is a convex pentagon such that A B = B C, C D = D E, M is middle point of side EA and the angles A B C ^ = C D E ^ = 90 ° , find the measure of the angle B M D ^ .

Question

Pentagon with two right angles (aka Van Aubel&#039;s Theorem)Given that ABCDE is a convex pentagon such that   A  B  =  B  C,   C  D  =  D  E, M is middle point of side EA and the angles                     A        B        C            ^        =                    C        D        E            ^        =  90      °  , find the measure of the angle                     B        M        D            ^      .

Reese Hobbs · Accepted Answer

Step 1It is a right angle.This can be stated in the following way: if ABC is a triangle,       O    B    ,      O    C  , are the center of the squares (externally) built on AC and AB and M is the midpoint of BC, then   M      O    B    ⊥  M      O    C  .I believe this is also known as Van Aubel&#039;s theorem.Proof: embed the construction in       C   and assume without loss of generality   B  =  −  1  ,  C  =  1.We have   M  =  0 and       O    C    =  (  A  −  B  )  ⋅            1      +      i        2    +  B and       O    B    =  (  A  −  C  )  ⋅            1      −      i        2    +  C, so:      O    C    =            1      +      i        2    A  +            i      −      1        2    ,        O    B    =            1      −      i        2    A  +            1      +      i        2   and:       O    C    =  i  ⋅      O    B    ..Step 2This proves   M      O    C    ⊥  M      O    B   and   M      O    B    =  M      O    C    ,, too.

Izabelle Lowery · Accepted Answer

Step 1Let   C  =      0  , put   A  =  2      a  ,   E  =  2      e  , and denote by   ∗      x   the vector x turned       90    ∘   counterclockwise. Then   M  =      a    +      e    ,    D  =      e    −  ∗      e    ,    B  =      a    +  ∗      a       .It follows that             B      M        →    =  (      a    +      e    )  −  (      a    +  ∗      a    )  =      e    −  ∗      a   and then             D      M        →    =  (      a    +      e    )  −  (      e    −  ∗      e    )  =      a    +  ∗      e    =  ∗  (      e    −  ∗      a    )  =  ∗            B      M        →       ,, which shows that       |    B  M      |    =      |    D  M      |   as well.

Given that ABCDE is a convex pentagon such that AB=BC, CD=DE, M is middle point of side EA and the angles hat{ABC}=hat{CDE}=90^circ, find the measure of the angle hat{BMD}.

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