3

Part of 1992 Bundeswettbewerb Mathematik

Problems(2)

3 spheres touch plane of a triangle at it's vertices, tangent in pairs

Source: 1992 German Federal - Bundeswettbewerb Mathematik - BWM - Round 1 p3

Given is a triangle

ABC

with side lengths

a, b,c

. Three spheres touch each other in pairs and also touch the plane of the triangle at points

A,B

C

, respectively. Determine the radii of these spheres.

geometry3D geometrysphere

convex equilateral pentagon

Source: Bundeswettbewerb Mathematik 1992, round two, problem 3

Provided a convex equilateral pentagon. On every side of the pentagon We construct equilateral triangles which run through the interior of the pentagon. Prove that at least one of the triangles does not protrude the pentagon's boundary.

trigonometrygeometry unsolvedgeometry