Question

If a vertex of a triangle is `(1,1)` , and the middle points of two sides passing through it are `-2,3)` and `(5,2),` then find the centroid and the incenter of the triangle.

Answer 1

Let E and F be the midpoints of AB and AC. Let the coordinates of B and C be `(alpha,beta)` and `(gamma, delta)` respectively. Then `-2=(1+alpha)/(2),3=(1+beta)/(2)`,
`5=(1+gamma)/(2),2=(1+delta)/(2)`
`therefore alpha=-5,beta=5,gamma=9,delta=3`

Therefore, the coordinates of B and C are `(-5,5)` and `(9,3)`, respectively.
Then the centroid is
`((1-5+9)/(3),(1+5+9)/(3))-=((5)/(3),3)`
and `a=BC=sqrt((-5-9)^2+(5-3)^2)=10sqrt(2)`
`b=CA=sqrt((9-1)^2+(3-1)^2)=2sqrt(17)`
`c=AB=sqrt((1+5)^2+(1-5)^2)=2sqrt(13)`
Then, the incenter is
`((10sqrt2(1)+2sqrt(17)(-5)+2sqrt13(9))/(10sqrt2+2sqrt17+2sqrt13),(10sqrt2(1)+2sqrt(17)(5)+2sqrt13(3))/(10sqrt2+2sqrt17+2sqrt13))`
i.e., `((5sqrt2-5sqrt(17)+9sqrt13)/(5sqrt2+sqrt17+sqrt13),(5sqrt2+5sqrt(17)+3sqrt13)/(5sqrt2+sqrt17+sqrt13))`