Fix the triangle ABC on the plane.
1. Denote by SL,SM and SK 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 SK≤SL≤SM.2. For the point X, which is inside the triangle ABC, consider the triangle TX, 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 Tx is the largest possible.
2.2. Suggest an effective criterion for comparing the areas of triangles Tx for different positions of the point X.
2.3. Find the positions of the point X for which the perimeter of the triangle Tx is the smallest possible and the largest possible.
2.4. Propose an effective criterion for comparing the perimeters of triangles Tx 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 Tx.3. For the point Y, which is inside the circle ω, circumscribed around the triangle ABC, consider the triangle ΔY, the vertices of which are the points of intersection AY,BX,CX with the circle ω. Suggest and solve similar problems for triangles ΔY 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 ω, circumscribed around the triangle ABC, consider the triangle FZ, 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 FZ for different positions of the point Z. geometrygeometric inequalityUkrainian TYM