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Q. ABC is an equilateral triangle with side length 6. If X is a point on side CA , what is the minimum value of BX² + XC² ?
Answer: Let's D is mid point of CA (BD = height = 3√3)
So, Bx^2 + Cx^2 = (3√3)^2 + Dx^2 + Cx^2 = 27 + Dx^2 + Cx^2
So minima = 27 + 2Dx.Cx (from AM GM)
Now (Dx + Cx) = 3
DX^2 +CX^2 = (9-2DX.CX)
2Dx.Cx = 9-2Dx.Cx
2Dx.Cx = 9/2
Ans: 27+9/2 = 63/2
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