MathDB

3

Part of 2006 BAMO

Problems(1)

2006 BAMO p3 P is centroid of ABC iff P is centroid of A_1B_1C_1

Source:

8/27/2019
In triangle ABCABC, choose point A1A_1 on side BCBC, point B1B_1 on side CACA, and point C1C_1 on side ABAB in such a way that the three segments AA1,BB1AA_1, BB_1, and CC1CC_1 intersect in one point PP. Prove that PP is the centroid of triangle ABCABC if and only if PP is the centroid of triangle A1B1C1A_1B_1C_1.
Note: A median in a triangle is a segment connecting a vertex of the triangle with the midpoint of the opposite side. The centroid of a triangle is the intersection point of the three medians of the triangle. The centroid of a triangle is also known by the names ”center of mass” and ”medicenter” of the triangle.
Centroidmediangeometry