Given an equilateral triangle $ABC$, where points $P$ and $Q$ divide the side $BC$ in ratios 1:1 and 1:2, respectively, and $R$ is the midpoint of $AP$, we can analyze the situation as follows:

Step 1: Coordinate Placement

Let's place the triangle $ABC$ in a coordinate system:

• Let $A = (0, \sqrt{3}a)$,
• Let $B = (-a, 0)$,
• Let $C = (a, 0)$.

Step 2: Determine Coordinates of Points $P$ and $Q$

• Point $P$: Since $P$ divides $BC$ in a $1:1$ ratio, $P$ is the midpoint of $BC$.

  • The coordinates of $P$ are calculated as: $P = \left(\frac{-a + a}{2}, \frac{0 + 0}{2}\right) = (0, 0)$

• Point $Q$: $Q$ divides $BC$ in a $1:2$ ratio, so: $Q = \left(\frac{2(-a) + a}{3}, \frac{2(0) + 0}{3}\right) = \left(-\frac{a}{3}, 0\right)$

Step 3: Determine the Coordinates of Point $R$

Since $R$ is the midpoint of $AP$:

• $R$ is the midpoint of $A(0, \sqrt{3}a)$ and $P(0, 0)$.
  • The coordinates of $R$ are: $R = \left(\frac{0 + 0}{2}, \frac{\sqrt{3}a + 0}{2}\right) = \left(0, \frac{\sqrt{3}a}{2}\right)$

Summary

• The coordinates of $P$ are $(0, 0)$,
• The coordinates of $Q$ are $\left(-\frac{a}{3}, 0\right)$,
• The coordinates of $R$ are $\left(0, \frac{\sqrt{3}a}{2}\right)$.

If you need further details or have any questions, feel free to ask!

Here are some related questions:

1. What are the properties of the centroid in an equilateral triangle?
2. How do you find the circumcenter of an equilateral triangle?
3. What is the significance of the centroid in relation to the sides of a triangle?
4. How can we use vectors to solve this problem?
5. What are the general properties of an equilateral triangle?

Tip: Always double-check the calculations, especially when dealing with coordinate geometry, to ensure the accuracy of the points and their relationships.