smc

First, find $P$, $K$, and $R$ in terms of $A$. Since all sides of an equilateral triangle are the same, $P=3A$. From the area formula, $K=\frac{A^2\sqrt{3}}{4}$. By using 30-60-90 triangles, $R=\frac{A\sqrt{3}}{3}$. Using the same steps, $p=3a$, $k=\frac{a^2\sqrt{3}}{4}$, and $r=\frac{a\sqrt{3}}{3}$. Note that $P/p=3A/3a=A/a$ and $R/r=\frac{A\sqrt{3}}{3}÷\frac{a\sqrt{3}}{3}=A/a$. That means $P/p=R/r$, so the answer is $\boxed{P:p=R:r\text{always}}$.