# Geometry Proofs : Triangles, Bisectors and Midpoints

1- Given triangle as shown (infigure 241W) with AF = 1/4 AB, AE = 1/3 and BE Intersect CF = O (BE has ray bar on top and CF has distance bar on top) prove that AO hits BC At its midpoint.

2- Use Ceva's theorem to prove that the internal bisectors of the angles of a triangle are concurrent

(cevas theorem: given triangle ABC and 3 cevian lines L1, L2, L3 for each vertex hitting the opposite side at D, E,F respectively then L1,L2,L3 are concurren <=> AF/FB * BD/DC * CE/EA = 1)

3- Suppose that A', B', C' are midpoints of the sides of a triangle ABC and are also midpoints of KK' subset BC, LL' subset AC, MM' subset AB. Show that if AK', BL', CM' are also concurrent. (figure 241W-5)

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1. Let AO intersect BC at M, then we need to prove that M is the center of BC.

Connect EF, and let EF intersect AO at N,

We know, AE = 1/3 EC so AE/AC = ¼ = AF/AB

So we know EF is parallel to BC. So we get ...

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In this solution we investigate Triangles, Bisectors and Midpoints. The solution is detailed and well presented.