MathDB

2

Part of 2005 Italy TST

Problems(2)

Distances between centroid and the sides of a triangle

Source:

11/9/2010
(a)(a) Prove that in a triangle the sum of the distances from the centroid to the sides is not less than three times the inradius, and find the cases of equality. (b)(b) Determine the points in a triangle that minimize the sum of the distances to the sides.
geometryinradiusinequalitiesrearrangement inequalitygeometry proposed
F,C,H are collinear

Source:

11/9/2010
The circle Γ\Gamma and the line ℓ\ell have no common points. Let ABAB be the diameter of Γ\Gamma perpendicular to ℓ\ell, with BB closer to ℓ\ell than AA. An arbitrary point C≠AC\not= A, BB is chosen on Γ\Gamma. The line ACAC intersects ℓ\ell at DD. The line DEDE is tangent to Γ\Gamma at EE, with BB and EE on the same side of ACAC. Let BEBE intersect ℓ\ell at FF, and let AFAF intersect Γ\Gamma at G≠AG\not= A. Let HH be the reflection of GG in ABAB. Show that F,CF,C, and HH are collinear.
geometrygeometric transformationreflectionpower of a pointgeometry unsolved