Question

Given that point G is the center of gravity of triangle ABC, prove the identity AG⁽²⁾+BG⁽²⁾+CG⁽²⁾=(1)/(3)(AB⁽²⁾+BC⁽²⁾+CA⁽²⁾)

Answer 1

To prove the identity AG²+BG²+CG²=(1/3)(AB²+BC²+CA²), we can use the property that the center of gravity of a triangle divides the medians into a 2:1 ratio.

Step-by-step explanation:

In order to prove the identity AG⁹²+BG⁹²+CG⁹²=∅ABC⁹²+BC⁹²+CA⁹²

We can use the property that the center of gravity of a triangle divides the medians into a 2:1 ratio.

1. Let D, E, and F be the midpoints of BC, AC, and AB respectively.
2. Then, we can say that GD = 2/3 * AD, GE = 2/3 * BE, and GF = 2/3 * CF.
3. Now, using the distance formula, we can find the lengths of AG, BG, and CG.
4. Substituting these values into the left-hand side of the identity and simplifying, we can prove the given identity.