MathDB

5

Part of 2012 Sharygin Geometry Olympiad

Problems(4)

Tangent and parallel lines

Source: Sharygin Geometry Olympiad 2012 - Problem 5

4/28/2012
On side ACAC of triangle ABCABC an arbitrary point is selected DD. The tangent in DD to the circumcircle of triangle BDCBDC meets ABAB in point C1C_{1}; point A1A_{1} is defined similarly. Prove that A1C1∥ACA_{1}C_{1}\parallel AC.
geometrycircumcircleratiocyclic quadrilateralgeometry unsolved
sum of distances from interior point P from vertices of ABCD > it's perimeter

Source: 2012 Sharygin Geometry Olympiad Final Round 8.5

8/3/2018
Do there exist a convex quadrilateral and a point PP inside it such that the sum of distances from PP to the vertices of the quadrilateral is greater than its perimeter?
(A.Akopyan)
geometryperimeterconvex polygon
angle chasing starting with an isosceles right triangle.

Source: 2012 Sharygin Geometry Olympiad Final Round 9.5

8/3/2018
Let ABCABC be an isosceles right-angled triangle. Point DD is chosen on the prolongation of the hypothenuse ABAB beyond point AA so that AB=2ADAB = 2AD. Points MM and NN on side ACAC satisfy the relation AM=NCAM = NC. Point KK is chosen on the prolongation of CBCB beyond point BB so that CN=BKCN = BK. Determine the angle between lines NKNK and DMDM.
(M.Kungozhin)
Angle Chasingisoscelesright trianglegeometry
maximal radii of circles inscribed into 2 crescents are equal

Source: 2012 Sharygin Geometry Olympiad Final Round 10.5

8/3/2018
A quadrilateral ABCDABCD with perpendicular diagonals is inscribed into a circle ω\omega. Two arcs α\alpha and β\beta with diameters AB and CDCD lie outside ω\omega. Consider two crescents formed by the circle ω\omega and the arcs α\alpha and β\beta (see Figure). Prove that the maximal radii of the circles inscribed into these crescents are equal.
(F.Nilov)
geometryarccirclesinscribed circles