MathDB

2

Part of 2012 Sharygin Geometry Olympiad

Problems(4)

Similar triangles in n-gon

Source: Sharygin Geometry Olympiad 2012 - Problem 2

4/28/2012
A cyclic nn-gon is divided by non-intersecting (inside the nn-gon) diagonals to n−2n-2 triangles. Each of these triangles is similar to at least one of the remaining ones. For what nn this is possible?
geometrycircumcirclegeometry unsolved
triangle construction given cuts of <B,<C bisectors with sides + point A

Source: 2012 Sharygin Geometry Olympiad Final Round 8.2

8/3/2018
In a triangle ABCABC the bisectors BB′BB' and CC′CC' are drawn. After that, the whole picture except the points A,B′A, B', and C′C' is erased. Restore the triangle using a compass and a ruler.
(A.Karlyuchenko)
geometryangle bisectorconstruction
3 parallel lines intersecting circumecirle, reflections, and lead to concurrent

Source: 2012 Sharygin Geometry Olympiad Final Round 9.2

8/3/2018
Three parallel lines passing through the vertices A,BA, B, and CC of triangle ABCABC meet its circumcircle again at points A1,B1A_1, B_1, and C1C_1 respectively. Points A2,B2A_2, B_2, and C2C_2 are the reflections of points A1,B1A_1, B_1, and C1C_1 in BC,CABC, CA, and ABAB respectively. Prove that the lines AA2,BB2,CC2AA_2, BB_2, CC_2 are concurrent.
(D.Shvetsov, A.Zaslavsky)
geometrygeometric transformationreflectionconcurrencyconcurrent
lengths of the cevians passing through P are inversely proportional sidelengths

Source: 2012 Sharygin Geometry Olympiad Final Round 10.2

8/3/2018
We say that a point inside a triangle is good if the lengths of the cevians passing through this point are inversely proportional to the respective side lengths. Find all the triangles for which the number of good points is maximal.
(A.Zaslavsky, B.Frenkin)
geometryratiosCevian