Let △ A B C be a scalene triangle, I its incenter and G its centroid. Prove that the line GI intersects both (AB) and (AC) on the line segments (id est between the two points of extremities), if and only if there exists a t ∈ [ 0 ; 1 ) for which we may write a = t b + ( 1 − t ) c , where a, b, c are the sides of the triangle in trigonometric notation (id est a = B C , b = A C , c = A B ).

Question

Let   △  A  B  C be a scalene triangle, I its incenter and G its centroid. Prove that the line GI intersects both (AB) and (AC) on the line segments (id est between the two points of extremities), if and only if there exists a   t  ∈      [    0    ;    1    )   for which we may write   a  =  t  b  +  (  1  −  t  )  c , where a, b, c are the sides of the triangle in trigonometric notation (id est   a  =  B  C ,   b  =  A  C ,   c  =  A  B ).

candelo6a · Accepted Answer

Step 1The barycentric coordinates of G with respect to ABC are (1, 1, 1), and those of I are (a, b, c). If GI intersects BC at a point with barycentric coordinates (0, y, z), we get the equation      |                            1                          1                          1                                      a                          b                          c                                      0                          y                          z                      |    =  0.Hence   (  b  −  a  )  z  =  (  c  −  a  )  y . Line GI intersects line BC outside segment BC if and only if the ratio of y to z is negative, which occurs precisely when a is the middle length.Analogous statements can be made for lines AB and AC.

Leonel Contreras · Accepted Answer

Step 1  △  A  B  C is a scalene triangle. Now here is a geometrical argument.WLOG we assume that   c  &amp;gt;  b and then we need to show that   c  &amp;gt;  a  &amp;gt;  bFor the line l through G and I to cut the segment AC internally, incenter I must be above median CD or in other words,                     A        H                    B        H              &amp;lt;    1   .But given CH is an angle bisector,                     A        H                    B        H              =          b      a            ⟹        b    &amp;lt;    a   .Now for the line l to cut the segment AB internally, incenter I must be below median BE, so                     C        K                    A        K              =          a      c        &amp;lt;    1        ⟹        a    &amp;lt;    c  .

Let <mi mathvariant="normal">△ A B C be a scalene triangle, I its in

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