Three cevians divided the triangle into six triangles, the areas of which are marked in the figure. 1) Prove that $S_1 \cdot S_2 \cdot S_3 =Q_1 \cdot Q_2 \cdot Q_3$. 2) Determine whether it is true that if $S_1 = S_2 = S_3$, then $Q_1 = Q_2 = Q_3$.

Three cevians are drawn in a triangle that do not intersect at one point. In this case, $4$ triangles and $3$ quadrangles were formed. Find the sum of the areas of the quadrilaterals if the area of each of the four triangles is $8$.

In the equilateral triangle $ABC$ ($AB = 2$), cevians are drawn that do not intersect at one point. It turned out that the pairwise intersection points of these cevians lie on the inscribed circle of triangle $ABC$. Find the length of the cevian segment from the vertex of the triangle to the nearest point of intersection with the circle.