Square of the angle bisector in a triangle.Prove that the square of the length, 'd' of the angle bisector AL of ∠ A meeting BC at L in a Δ A B C is b c ( 1 − ( a b + c ) 2 ) .

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Square of the angle bisector in a triangle.Prove that the square of the length, &#039;d&#039; of the angle bisector AL of   ∠  A meeting BC at L in a   Δ  A  B  C is   b  c      (    1    −                  (                  a                      b            +            c                          )            2        )    .

Alannah Yang · Accepted Answer

Step 1By the Stewart&#039;s theorem                               c          2                u        +                  b          2                (        a        −        u        )                            =        a        (                  d          2                +        (        a        −        u        )        u        )        ,                            (1)                              d          2                                    =                  1          a                        (                  c          2                u        +                  b          2                (        a        −        u        )        )        −        (        a        −        u        )        u        ,            Step 2by the angle bisector theorem,                     (2)                    b                (        a        −        u        )                            =        c                u        ,                            (3)                    u                            =                              a            b                                b            +            c                          ,                            (4)                    a        −        u                            =                              a            c                                b            +            c                          .            Combination of (1) with (3) and (4) gives                               d          2                                    =                              b            c            (            (            b            +            c                          )              2                        −                          a              2                        )                                (            b            +            c                          )              2                                      =        b        c                  (          1          −                                    a              2                                      (              b              +              c                              )                2                                              )                .

Prove that the square of the length, 'd' of the angle bisector AL of angle A meeting BC at L in a triangle ABC is bc(1-(a/(a+c))^2).

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