Maximizing the Area of Triangle BDF

Problem

As shown in the figure, the triangle $ABC$ has area $2$ . Points $D$ and $E$ lie on the sides $AB$ and $AC$ respectively, and $F$ is a point on the segment $DE$ . Let

x = \dfrac{AD}{AB}, \quad y = \dfrac{AE}{AC}, \quad z = \dfrac{DF}{DE},

and suppose $y + z - x = 1$ . Find the maximum possible area of $\triangle BDF$ .

Answer

Solution

Difficulty	6/10
Topics	plane geometry, AM-GM, inequality

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