In A B C, ∠ B &lt; ∠ C . If D and E are in A C and A B respectively, such that B D and C E are angle bisectors, prove that B D &gt; C E.Well, A C &lt; A B, but is there a formula for angle bisector length?

Question

In   A  B  C,   ∠      B    &amp;lt;  ∠      C  . If   D and   E are in   A  C and   A  B respectively, such that   B  D and   C  E are angle bisectors, prove that   B  D  &amp;gt;  C  E.Well,   A  C  &amp;lt;  A  B, but is there a formula for angle bisector length?

Eleanor Luna · Accepted Answer

Geometry of a triangle is studied extensively. Let the sides be   a,  b,  c respectively and       l    a   be the length of angle bisector from vertex   A onto the side whose length is   a  . The Angle Bisector property tells us that       l    a   bisects the side   B  C in the proportion   A  C  :  A  B  =  b  :  c  . This gives us two equations:      {                            x          +          y          =          a                                                                x              y                                =                                    b              c                                              Solving this is easy and it yields:  x  =                    a        b                    b        +        c                ,  y  =                    a        c                    b        +        c              .Finally, Stewart&#039;s theorem gives us:  a  ⋅      l    a    2    =                              b          2                a        c        +                  c          2                a        b                    b        +        c              −                              a          3                b        c                    (        b        +        c                  )          2                    or      l    a    =      b    c    −                                        a            2                    b          c                          (          b          +          c                      )            2                                =                    2                  b          c          p          (          p          −          a          )                            b        +        c            where   p  =                    a        +        b        +        c            2       is the semiperimeter.With this formula in hand and in the setup of your problem, one can immediately see that:  C      E    2    −  B      D    2    =      l    c    2    −      l    b    2    =  a  b  −                              c          2                a        b                    (        a        +        b                  )          2                      −  a  c  +                              b          2                a        c                    (        a        +        c                  )          2                      =  =  a  (  b  −  c  )      (    1    +                            b          c          (                      a            2                    +                      b            2                    +                      c            2                    +          b          c          +          2          a          b          +          2          a          c          )                          (          a          +          b                      )            2                    (          a          +          c                      )            2                                )    ≤  0.Thus,       l    b    ≥      l    c    .

In A B C , <mi mathvariant="normal">∠ <mrow class="MJX-TeXAtom-OR

Answered question

Answer & Explanation

New Questions in Elementary geometry