I'm taking a course at teaching and we have some geometry questions. Among the questions there was one I couldn't solve.I'm trying to prove that angle bisectors in a triangle intersect at a single point, but I need to show it analytically, i.e using line equations and distance between points. I'm familiar with proofs using geometry and vectors, but couldn't manage to show analytically. I can't use the formula of angle between lines or distance of point from a given line.Things I've tried:1. angle bisector theorem.2. Writing equations of sides as A x + B y + C = 0 and then creating the equation of angle bisector - quite messy and I believe there is a lot easier way.Thanks!

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I&#039;m taking a course at teaching and we have some geometry questions. Among the questions there was one I couldn&#039;t solve.I&#039;m trying to prove that angle bisectors in a triangle intersect at a single point, but I need to show it analytically, i.e using line equations and distance between points. I&#039;m familiar with proofs using geometry and vectors, but couldn&#039;t manage to show analytically. I can&#039;t use the formula of angle between lines or distance of point from a given line.Things I&#039;ve tried:1. angle bisector theorem.2. Writing equations of sides as   A  x  +  B  y  +  C  =  0 and then creating the equation of angle bisector - quite messy and I believe there is a lot easier way.Thanks!

Halle Mckee · Accepted Answer

Without loss of generality, we can set:   A  =  (  0  ,  0  ),   B  =  (  1  ,  0  ) and   C  =  (  a  ,  b  ), with   b  &amp;gt;  0. By the tangent bisection formula one gets then the slopes   m and       m    ′   of the angle bisectors through   A and   B:  m  =                              1          −          a                      /                                              a              2                        +                          b              2                                                1          +          a                      /                                              a              2                        +                          b              2                                            ,        m    ′    =  −                              1          −          (          1          −          a          )                      /                                (            1            −            a                          )              2                        +                          b              2                                                1          +          (          1          −          a          )                      /                                (            1            −            a                          )              2                        +                          b              2                                            .You can now find the equations of the two lines and their intersection   G. It remains to show that   C  G is the bisector of angle   C. I think the simplest way to do that is finding the intersection   H between line   C  G and the   x axis, and checking that   A  H      /    B  H  =  A  C      /    B  C.

I'm taking a course at teaching and we have some geometry questions. Among the questions there was o

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