Given A = ( a 1 , a 2 ) , B = ( b 1 , b 2 ) , C = ( c 1 , c 2 ), is there a simple formula to express the radius of the circumcircle of ABC?Note that you could compute the radius from the sidelengths as a b c ( a + b + c ) ( − a + b + c ) ( a − b + c ) ( a + b − c ) , but I'm really hoping that there's something simpler than that. If it helps, assume C=(0,0).

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Given   A  =  (      a    1    ,      a    2    )  ,  B  =  (      b    1    ,      b    2    )  ,  C  =  (      c    1    ,      c    2    ), is there a simple formula to express the radius of the circumcircle of ABC?Note that you could compute the radius from the sidelengths as             a      b      c              (      a      +      b      +      c      )      (      −      a      +      b      +      c      )      (      a      −      b      +      c      )      (      a      +      b      −      c      )      , but I&#039;m really hoping that there&#039;s something simpler than that. If it helps, assume C=(0,0).

Cindy Mercer · Accepted Answer

The square root in the denominater is equal to the twice the area of   △  A  B  C (Heron&#039;s formula). The same area can be calculated by embeding the plane in 3-dimensional space and using the vector product:  S  =      1    2        |                      C        A            →        ×                    C        B            →            |  Using Cartesian coordinates, we have      |                      C        A            →        ×                    C        B            →            |    =      |    (      a    1    −      c    1    )  (      b    2    −      c    2    )  −  (      a    2    −      c    2    )  (      b    1    −      c    1    )      |    =    =      |        a    1    (      b    2    −      c    2    )  +      b    1    (      c    2    −      a    2    )  +      c    1    (      a    2    −      b    2    )      |  so  R  =                    (                  a          1                −                  c          1                          )          2                +        (                  a          2                −                  c          2                          )          2                            (                  b          1                −                  c          1                          )          2                +        (                  b          2                −                  c          2                          )          2                            (                  a          1                −                  b          1                          )          2                +        (                  a          2                −                  b          2                          )          2                                    |                    a        1            (              b        2            −              c        2            )      +              b        1            (              c        2            −              a        2            )      +              c        1            (              a        2            −              b        2            )              |

Given A=(a_1,a_2),B=(b_1,b_2),C=(c_1,c_2), is there a simple formula to express the radius of the circumcircle of ABC?

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