MathDB

2

Part of 1998 Hungary-Israel Binational

Problems(2)

inequality on radius

Source: 9th-th Hungary-Israel Binational Mathematical Competition 1998

7/13/2007
A triangle ABC is inscribed in a circle with center O O and radius R R. If the inradii of the triangles OBC,OCA,OAB OBC, OCA, OAB are r1,r2,r3 r_{1}, r_{2}, r_{3} , respectively, prove that 1r1+1r2+1r3≥43+6R. \frac{1}{r_{1}}+\frac{1}{r_{2}}+\frac{1}{r_{3}}\geq\frac{4\sqrt{3}+6}{R}.
inequalitiestrigonometrygeometrycircumcirclegeometry proposed
triangles are constructd in exterior of a convex hexagon

Source: 9th-th Hungary-Israel Binational Mathematical Competition 1998

7/13/2007
On the sides of a convex hexagon ABCDEF ABCDEF , equilateral triangles are constructd in its exterior. Prove that the third vertices of these six triangles are vertices of a regular hexagon if and only if the initial hexagon is affine regular. (A hexagon is called affine regular if it is centrally symmetric and any two opposite sides are parallel to the diagonal determine by the remaining two vertices.)
geometryparallelogramgeometry proposed