MathDB

4

Part of 2014 Sharygin Geometry Olympiad

Problems(4)

Incenter Lies Inside Inscribed Square

Source: Sharygin Geometry Olympiad 2014 - Problem 4

11/15/2014
A square is inscribed into a triangle (one side of the triangle contains two vertices and each of two remaining sides contains one vertex. Prove that the incenter of the triangle lies inside the square.
geometryincentergeometry unsolved
constructing a triangle by paperfolding, original grid faded, sidelengths known

Source: 2014 Sharygin Geometry Olympiad Final Round 8.4

8/3/2018
Tanya has cut out a triangle from checkered paper as shown in the picture. The lines of the grid have faded. Can Tanya restore them without any instruments only folding the triangle (she remembers the triangle sidelengths)?
(T. Kazitsyna)
geometryconstruction
3 circles with centers A, B, C and radii AH, BH, CH have a common tangent

Source: 2014 Sharygin Geometry Olympiad Final Round 9.4

8/3/2018
Let HH be the orthocenter of a triangle ABCABC. Given that HH lies on the incircle of ABCABC , prove that three circles with centers A,B,CA, B, C and radii AH,BH,CHAH, BH, CH have a common tangent.
(Mahdi Etesami Fard)
common tangentscirclesgeometry
a point is good if 6 points are concyclic

Source: 2014 Sharygin Geometry Olympiad Final Round 10.4

8/3/2018
Let ABCABC be a fixed triangle in the plane. Let DD be an arbitrary point in the plane. The circle with center DD, passing through AA, meets ABAB and ACAC again at points AbA_b and AcA_c respectively. Points Ba,Bc,CaB_a, B_c, C_a and CbC_b are defined similarly. A point DD is called good if the points Ab,Ac,Ba,Bc,CaA_b, A_c,B_a, B_c, C_a, and CbC_b are concyclic. For a given triangle ABCABC, how many good points can there be?
(A. Garkavyj, A. Sokolov )
geometryConcyclic