Fix the triangle $ABC$ on the plane. 1. Denote by $S_L,S_M$ and $S_K$ the areas of triangles whose vertices are, respectively, the bases of bisectors, medians and points of tangency of the inscribed circle of a given triangle $ABC$. Prove that $S_K\le S_L\le S_M$. 2. For the point $X$, which is inside the triangle $ABC$, consider the triangle $T_X$, the vertices of which are the points of intersection of the lines $AX, BX, CX$ with the lines $BC, AC, AB$, respectively. 2.1. Find the position of the point $X$ for which the area of the triangle $T_x$ is the largest possible. 2.2. Suggest an effective criterion for comparing the areas of triangles $T_x$ for different positions of the point $X$. 2.3. Find the positions of the point $X$ for which the perimeter of the triangle $T_x$ is the smallest possible and the largest possible. 2.4. Propose an effective criterion for comparing the perimeters of triangles $T_x$ for different positions of point $X$. 2.5. Suggest and solve similar problems with respect to the extreme values of other parameters (for example, the radius of the circumscribed circle, the length of the greatest height) of triangles $T_x$. 3. For the point $Y$, which is inside the circle $\omega$, circumscribed around the triangle $ABC$, consider the triangle $\Delta_Y$, the vertices of which are the points of intersection $AY, BX, CX$ with the circle $\omega$. Suggest and solve similar problems for triangles $\Delta_Y$ for different positions of point $Y$. 4. Suggest and solve similar problems for convex polygons. 5. For the point $Z$, which is inside the circle $\omega$, circumscribed around the triangle $ABC$, consider the triangle $F_Z$, the vertices of which are orthogonal projections of the point $Z$ on the lines $BC$, $AC$ and $AB$. Suggest and solve similar problems for triangles $F_Z$ for different positions of the point $Z$.