P is a point on the circumference of an incircle of an equilateral triangle ABC with side 2 cm each. Find (AP)²+(BP)²+(CP)²

Given an equilateral triangle \(ABC\) with side length \(2 \, \text{cm}\), and a point \(P\) on the circumference of its incircle, we need to find the value of \((AP)^2 + (BP)^2 + (CP)^2\).

### Key Concepts:
1. **Centroid of Equilateral Triangle:** 
   The centroid \(G\) of an equilateral triangle coincides with the center of the incircle and the circumcircle.
   
2. **Distances from any point on the Incircle:**
   For any point \(P\) on the incircle of an equilateral triangle \(ABC\) with side length \(a\), it holds that:
   \[
   (AP)^2 + (BP)^2 + (CP)^2 = 3r^2 + \frac{a^2}{2}
   \]
   where \(r\) is the radius of the incircle.

### Calculation:

#### Step 1: Find the Radius \(r\) of the Incircle
For an equilateral triangle, the radius \(r\) of the incircle is given by:
\[
r = \frac{a \sqrt{3}}{6}
\]
where \(a\) is the side length of the triangle. Substituting \(a = 2 \, \text{cm}\):
\[
r = \frac{2 \sqrt{3}}{6} = \frac{\sqrt{3}}{3} \, \text{cm}
\]

#### Step 2: Use the Formula for \((AP)^2 + (BP)^2 + (CP)^2\)
Substituting the values:
\[
(AP)^2 + (BP)^2 + (CP)^2 = 3r^2 + \frac{a^2}{2}
\]
\[
= 3\left(\frac{\sqrt{3}}{3}\right)^2 + \frac{2^2}{2}
\]
\[
= 3 \cdot \frac{3}{9} + \frac{4}{2}
\]
\[
= 3 \cdot \frac{1}{3} + 2
\]
\[
= 1 + 2 = 3 \, \text{cm}^2
\]

Thus, the value of \((AP)^2 + (BP)^2 + (CP)^2\) is \( \boxed{3} \, \text{cm}^2 \).

Would you like further details on any step, or do you have any questions?

Here are some related questions you might explore:
1. How do we derive the formula for the radius of the incircle of an equilateral triangle?
2. What are the properties of the centroid of an equilateral triangle?
3. How does the location of point \(P\) affect the sum \((AP)^2 + (BP)^2 + (CP)^2\)?
4. Can this method be applied to non-equilateral triangles?
5. How does the formula change if \(P\) is on the circumcircle rather than the incircle?

**Tip:** For any regular polygon, the centroid and the center of the inscribed circle are the same point.

Learn how to calculate the sum of squares of distances from a point on the incircle of an equilateral triangle ABC with side length 2 cm. This problem involves properties of equilateral triangles, incircles, and circle geometry. Suitable for high school students.

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