Proving OG = 1/3(OA + OB + OC) in Triangles | Step-by-Step Explanation

[eekoz]

Okay, so I have a triagle with vertices A, B, and C.
I know that the centroid, G, is where all the medians of the triangle intersect, and G divides the median at a 2:1 ratio

Assuming point O is a point that's not on the triangle, how can I prove:
OG = 1/3(OA + OB + OC) ?

I've seen this equation a lot of times, but I'd like to see a proof of it for interest sake

Thanks

 

[AKG]

Let $P$ be the midpoint of [itex]\overline{BC}[/tex]. Then:

$$\overline{OB} + \frac{1}{2}\overline{BC} = \overline{OP}$$

$$\overline{OB} + \overline{BC} = \overline{OC}$$

$$\overline{OP} + \overline{PA} = \overline{OA}$$

$$\overline{OP} + \frac{1}{3}\overline{PA} = \overline{OG}$$

The last line comes from the fact that G divides the median $\overline{PA}$ at a 2:1 ratio. You should be able to figure it out from here.

 

[tacman]

for sharing this problem! Proving that OG = 1/3(OA + OB + OC) in a triangle can be done using the concept of medians and the properties of a centroid. Here's a step-by-step explanation:

Step 1: Draw a triangle ABC with vertices A, B, and C and label the centroid as G. Draw the medians from each vertex to the opposite side, creating six smaller triangles within the larger triangle.

Step 2: Let's label the points where the medians intersect the sides of the triangle as D, E, and F. These points divide each median into two segments, with G being the midpoint of each segment.

Step 3: Now, let's label the lengths of the sides of the triangle as a, b, and c, with a being opposite to A, b being opposite to B, and c being opposite to C.

Step 4: Since G is the centroid, we know that it divides each median into a 2:1 ratio. This means that GD = 2/3 of AD, GE = 2/3 of BE, and GF = 2/3 of CF.

Step 5: Using the properties of a median, we can also say that AD = b/2, BE = c/2, and CF = a/2.

Step 6: Substituting these values into our previous equations, we get GD = b/3, GE = c/3, and GF = a/3.

Step 7: Now, let's focus on triangle GDE. We can use the Pythagorean theorem to find the length of DE, which is the hypotenuse of this right triangle. DE^2 = GD^2 + GE^2. Substituting the values we found in step 6, we get DE^2 = (b/3)^2 + (c/3)^2.

Step 8: Using the same logic, we can find the length of DF in triangle GDF. DF^2 = GD^2 + GF^2. Substituting the values, we get DF^2 = (b/3)^2 + (a/3)^2.

Step 9: Similarly, in triangle GEF, we can find the length of EF. EF^2 = GE^2 + GF^2. Substituting the values, we get EF^2 = (c/3)^2