Geometry

1. Let ΔABC ∼ ΔA′B′C′. Prove that the following are equivalent:

(a) I = I′

(b) a\overrightarrow{AA′} + b\overrightarrow{BB′} + c\overrightarrow{CC′} = \overrightarrow{0}.

2. Let ΔABC and ΔA′B′C′. Prove that: 3\overrightarrow{GG′} = 2\overrightarrow{OO′} + \overrightarrow{HH′}.

3. Let ΔABC ∼ ΔA′B′C′ and I = I′, G = G′. If any two of the vectors \overrightarrow{AA′}, \overrightarrow{BB′}, \overrightarrow{CC′} aren’t collinear prove that ΔABC and ΔA′B′C′ are equilateral.

4. Let ΔABC and A′, B′, C′ the points in which the angle bisectors intersect the opposite sides. Prove that if ΔABC ∼ ΔA′B′C′ and I = I′ then ΔABC and ΔA′B′C′ are equilateral.

5. Let ABCD be a quadrilateral, {O} = AC ∩ BD and M a point all in the plane P. Consider the relation: 4\overrightarrow{MM′} = 2\overrightarrow{MO} + \overrightarrow{MA} + \overrightarrow{MB} + \overrightarrow{MC} + \overrightarrow{MD}.

(a) Prove that there exists the bijective function f : P → P so that f(M) = M′ for every M ∈ P.

(b) Prove that all lines MM′ pass trough a fixed point and determine it.

6. Prove that if the altitudes of a triangle can form another triangle than the initial triangle is equilateral.

7. Let ΔABC and M the midpoint of AB. Let M′ be its projection on the side AC and denote the centroid of ΔMM′A by G_{1}. Points G_{2}, G_{3} are obtained analogously. Prove that if ΔABC and ΔG_{1}G_{2}G_{3} have the same centroid then they are equilateral.

8. Let ΔABC and A′ the projection of A on BC. Let now M, N be the projections of A′ on AB, AC respectively. Denote the centroid of ΔAMN by G_{1}. Points G_{2}, G_{3} are obtained analogously. Prove that if ΔABC and ΔG_{1}G_{2}G_{3} have the same centroid then they are equilateral.

All problems are proposed by Călin Popa.