MathDB

4

Part of 2011 Sharygin Geometry Olympiad

Problems(4)

Bisectors of a triangle

Source: 2011 Sharygin Geometry Olympiad #4

5/22/2014
Segments AA′AA', BB′BB', and CC′CC' are the bisectrices of triangle ABCABC. It is known that these lines are also the bisectrices of triangle A′B′C′A'B'C'. Is it true that triangle ABCABC is regular?
geometrygeometric transformationreflectioninequalitiestriangle inequalitygeometry unsolved
circle of radius 1, chords with sum 1, inscribe reg. hexagon, no intersections

Source: Sharygin 2011 Final 8.4

12/13/2018
Given the circle of radius 11 and several its chords with the sum of lengths 11. Prove that one can be inscribe a regular hexagon into that circle so that its sides don’t intersect those chords.
circlegeometryregular polygonhexagoninscribedChordscombinatorics
3 cyclic quadrilaterals with equidistant centres

Source: Sharygin 2011 Round 2 grade 9 p4

7/2/2018
Quadrilateral ABCDABCD is inscribed into a circle with center OO. The bisectors of its angles form a cyclic quadrilateral with circumcenter II, and its external bisectors form a cyclic quadrilateral with circumcenter JJ. Prove that OO is the midpoint of IJIJ.
geometrycyclic quadrilateral
2 circles tangent internally to circumcircle, inscribed in <ADC, <BDC

Source: Sharygin 2011 Final 10.4

3/31/2019
Point DD lies on the side ABAB of triangle ABCABC. The circle inscribed in angle ADCADC touches internally the circumcircle of triangle ACDACD. Another circle inscribed in angle BDCBDC touches internally the circumcircle of triangle BCDBCD. These two circles touch segment CDCD in the same point XX. Prove that the perpendicular from XX to ABAB passes through the incenter of triangle ABCABC
geometrycircumcircletangent circlesperpendicularincenter