The point M is the internal point of the ABC equilateral triangle. Find the ∠ B M C if | M A | 2 = | M B | 2 + | M C | 2 .

Question

The point M is the internal point of the ABC equilateral triangle. Find the   ∠  B  M  C if       |    M  A            |        2    =      |    M  B            |        2    +      |    M  C            |        2  .

Brennan Parks · Accepted Answer

Step 1Use coordinate geometry. Note that       |    M  A            |        2    =      |    M  B            |        2    +      |    M  C            |        2   is symmetrical in B and C. This means M is located symmetrically with respect to B and C. Let the coordinates of A be   (  0  ,      3    a  ), B be (-a, 0) and C be (a,0). Since M has to be symmetrical, let the coordinates for M be (0,y). Step 2Now,       |    M  A            |        2    =      |    M  B            |        2    +      |    M  C            |        2      ⟹        y    2    +  2      3    a  y  −      a    2    =  0    ⟹    y  =  (  2  −      3    )  aNote that you&#039;ll get two values of y but it can&#039;t be negativeNow since you&#039;ve M, you can get the angle BMC to be equal to 150 degrees.

orkesruim40 · Accepted Answer

Step 1Rotate C around B for       60          ∘       and let N be a picture of M under this rotation. What can you say for triangles BMN and AMN?Step 2Triangle BMC must be equilateral and since   A      N    2    +  M      N    2    =  C      M    2    +  B      M    2    =  A      M    2   we see that triangle ANM is rectangular with 90 at N

The point M is the internal point of the ABC equilateral triangle. Find the angle BMC if |MA|^2=|MB|^2+|MC|^2.

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