# Thread: Synthetic, Analytic, & Vector Proof

1. ## Synthetic, Analytic, & Vector Proof

Hello All:

I have to prove the Centroid theorem in synthetic, analytic, and vector techniques. My problem is, I do not understand how to even begin or these techniques. I have looked through all of my old geometry books (I am a college student) and nothing is really in them about these techniques. Is there any easy way that this can be explained? I am completely lost and have to have this done by Feb. 1st and feeling like I am in too deep

Thanks for any assistance.

2. ## synthetic proof

Draw triangle ABC with midpoints F, D and E.
Using the crossbar theorem, show point G is the intersection between two angles.
Draw the midpoints of AG and CG.
Find similar triangles FBD and ABC; and AGC and hiG, then congruent triangles hDi, and FDh.
Draw parallelogram FhiD. The diagonal properties of a parallelogram state they bisect each other. If two different verteces were choosen instead of the original chosen, you will find the same intersection of G.
and because point G is inside each angle,G is in the interior of triangle
ABC, proving point G is the centroid.

Triangle Centroid -- from Wolfram MathWorld

In general, both the analytic, and vector techniques, while tedious, are rather straightforward.
The synthetic proof is the difficult one. Proving concurrence is rather hard. You need to know the ‘plane separation’ properties, definitions of interior points for both angles and triangles as well as theorems such as the ‘Cross Bar’ mentioned above.