MathDB

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Part of 2016 Oral Moscow Geometry Olympiad

Problems(2)

hexagon with angles, 3 main diagonals and other 6 diagonals, equal, regular?

Source: 2016 Oral Moscow Geometry Olympiad grades 8-9 p1

7/30/2019
Angles are equal in a hexagon, three main diagonals are equal and the other six diagonals are also equal. Is it true that it has equal sides?
hexagonanglesdiagonalsequal segmentsgeometry
point lies on a line passing through incenter and perpendicular to BC

Source: 2016 Oral Moscow Geometry Olympiad grades 10-11 p1

7/29/2019
The line passing through the center II of the inscribed circle of a triangle ABCABC, perpendicular to AIAI and intersects sides ABAB and ACAC at points C′C' and B′B', respectively. In the triangles BC′IBC'I and CB′ICB'I, the altitudes C′C1C'C_1 and B′B1B'B_1 were drawn, respectively. Prove that the midpoint of the segment B1C1B_1C_1 lies on a straight line passing through point II and perpendicular to BCBC.
geometryincenter