(E):x²/9+y²/4=1  M∈(E).F1(-√5;0).F2(√5;0). Find the incenter of triangle MF1F2.  

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`MF_2=f_1=sqrt((x-sqrt(5))^2+y^2)`

`MF_1=f_2=sqrt((x+sqrt(5))^2+y^2)` `x_(mn)`

`F_1F_2=m=sqrt((-sqrt(5)-sqrt(5))^2)=sqrt(20)`

let coordinate of incentre be `(barx,bary)`

so

`barx=(mx+f_1(-sqrt(5))+f_2sqrt(5))/(sqrt(20)+f_1+f_2)`

`barx=(sqrt(20)y-sqrt(5)sqrt((x-sqrt(5))^2+y^2)+sqrt(5)sqrt((x+sqrt(5))^2+y^2))/(sqrt(20)+sqrt((x-sqrt(5))^2+y^2)+sqrt((x+sqrt(5))^2+y^2))`

`bary=(my+f_1xx0+f_2xx0)/(sqrt(20)+f_1+f_2)`

`bary=(sqrt(20)y)/(sqrt(20)+sqrt((x-sqrt(5))^2+y^2)+sqrt((x+sqrt(5))^2+y^2)) `

`where 4x^2+9y^2=36 `

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