MathDB

2

Part of 2000 Iran MO (2nd round)

Problems(2)

ABC and DEF have the same centroid

Source:

10/15/2010
The points D,ED,E and FF are chosen on the sides BC,ACBC,AC and ABAB of triangle ABCABC, respectively. Prove that triangles ABCABC and DEFDEF have the same centroid if and only if BDDC=CEEA=AFFB\frac{BD}{DC} = \frac{CE}{EA}=\frac{AF}{FB}
geometryparallelogramsymmetrygeometry proposed
Faces of the tetrahedron are congruent triangle

Source:

10/15/2010
In a tetrahedron we know that sum of angles of all vertices is 180∘.180^\circ. (e.g. for vertex AA, we have ∠BAC+∠CAD+∠DAB=180∘.\angle BAC + \angle CAD + \angle DAB=180^\circ.) Prove that faces of this tetrahedron are four congruent triangles.
geometry3D geometrytetrahedrongeometry proposed