A Centroid Sitting on the Focus

Difficulty: Medium · Topics: Geometry, Conic Sections, Parabola

Fokusda joylashgan og‘irlik markazi

Three distinct points A, B, C lie on the parabola y^2=8x, and the centroid of triangle ABC coincides with the focus F of the parabola.

graph "x":[-1.2,5],"y":[-6,4.8],"xTicks":[],"yTicks":[],"label":"The parabola y² = 8x with three points A, B, C whose centroid is the focus F(2, 0)","items":["type":"parabola","p":2,"axis":"x","at":[0,0],"range":[-5.5,5.5],"type":"line","from":[2,0],"to":[2,4],"ink":true,"type":"line","from":[2,0],"to":[0.268,1.464],"ink":true,"type":"line","from":[2,0],"to":[3.732,-5.464],"ink":true,"type":"polygon","points":[[2,4],[0.268,1.464],[3.732,-5.464]],"ink":true,"dash":true,"type":"point","at":[2,4],"label":"A","anchor":"ne","type":"point","at":[0.268,1.464],"label":"B","anchor":"w","type":"point","at":[3.732,-5.464],"label":"C","anchor":"se","type":"point","at":[2,0],"label":"F","anchor":"se","ink":true]

Find the value of overlineAF+overlineBF+overlineCF.