# Mystic cevians 2

Geometry Level 5

Cevians $$AD, BE, CF$$ are constructed in a triangle $$ABC$$. Let the feet of perpendiculars on $$AD$$ from $$B$$ and $$C$$ respectively be $$B_{a}$$ and $$C_{a}$$, the feet of perpendiculars on $$BE$$ from $$C$$ and $$A$$ respectively be $$C_{b}$$ and $$A_{b}$$ and the feet of perpendiculars on $$CF$$ from $$A$$ and $$B$$ respectively be $$A_{c}$$ and $$B_{c}$$.

Suppose:

$$\large{\frac{BB_{a}}{CC_{a}}=\frac{CC_{b}}{AA_{b}}=\frac{AA_{c}}{BB_{c}}=\kappa}$$

It may be observed then, that the cevians shall be concurrent if and only if $$\kappa = 1$$. Suppose $$\kappa$$ differs from $$1$$ by $$\delta \%$$. Then the cevians shall meet pairwise at three different points $$P_{1},P_{2}$$ and $$P_{3}$$.

Determine for what value of $$\delta$$ will the triangle $$P_{1}P_{2}P_{3}$$ have an area $$\frac{\Delta}{2014}$$ where $$\Delta$$ is the area of triangle $$ABC$$.

$$\large{Note:}$$ Report the rounded off value for $$\delta$$, i.e. the integer closest to $$\delta$$. Inspired by a popular problem, otherwise, original!

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