Show that the area of triangle S A B C = R × M N

Question

Show that the area of ​triangle       S          A      B      C        =  R  ×  M  N

Amir Beck · Accepted Answer

Step 1First establish that   M  N  =  B  H  ×  sin  ⁡  ∠  A  B  C(law of sines -   sin  ⁡  (  ∠  H  M  B  )  :  B  H  =  sin  ⁡  (  ∠  A  B  C  )  :  M  N ).Then,  △  A  O  C is isosceles, let   ∠  O  A  C  =  ∠  O  C  A  =  β .Then   ∠  A  O  C  =      180    ∘    −  2  β  =      180    ∘    −  α  −  γ .where   α  ,  γ are the appropriate angles from the isosceles triangles   △  A  O  B and   △  B  O  C .The perpendicular from O through AC divides the angle in two, i.e.       90    ∘    −  α  −  γ , and so we arrive at   A  C  =  R  ×  sin  ⁡  (  ∠  A  B  C  ) .

Frederick Kramer · Accepted Answer

Step 1The orthocenter and the circumcenter of a triangle are isogonal conjugates, therefore  ∠  A  B  H  =  ∠  N  B  O  =      α      H  M  B  N  (  c  y  c  l  i  c  )    ⟹    ∠  B  H  M  =  ∠  M  N  B  =  90  −      α      ∴  ∠  B  F  N  =      90    0      ⟹    B  O  ⊥  M  N    [  B  M  O  N  ]  =  [  B  M  N  ]  −  [  M  N  O  ]  =            B      O      ⋅      M      N        2    =            R      ⋅      M      N        2    (  I  )    △  M  B  N  ∼  △  C  B  A  ∴            M      N              M      B        =      b    a          ∠    M  H  B  =  90  −  (  90  −      ∠    A  )    ⟹    M  B  =  B  H  s  e  n  (  ∠  A  )    [  B  M  O  N  ]  =            R      ⋅      b      a      ⋅      B      H      ⋅      s      e      n      (      ∠      A      )        2    =            [      A      B      C      ]      ⋅      R      s      e      n      (              ∠            A      )        a    =            [      A      B      C      ]        2    (  I  I  )    (  I  )  =  (  I  I  )  :            [      A      B      C      ]        2    =            R      .      M      N        2      ⟹                                [          A          B          C          ]          =          R          .          M          N

Show that the area of triangle S <mrow class="MJX-TeXAtom-ORD"> A B

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