Question 36
Three points $A'$, $B'$, $C'$ are located, respectively, on the sides $BC$, $CA$ and $AB$ of a given triangle $ABC$ of area $S$, so that $$
\frac{AC'}{AB} = \frac{BA'}{BC} = \frac{CB'}{CA} = p
$$ being $p$ a variable parameter, $0 \lt p \lt 1$. Determine 1) The area of the triangle $A'B'C'$ as a function of $p$. 2) The value of $p$ that minimizes the previous area. 3) The geometric place of the $P$ points of intersection of the parallels drawn by $A'$ and $C'$, respectively on the sides $AB$ and $AC$, when $p$ is between 0 and 1.
\frac{AC'}{AB} = \frac{BA'}{BC} = \frac{CB'}{CA} = p
$$ being $p$ a variable parameter, $0 \lt p \lt 1$. Determine 1) The area of the triangle $A'B'C'$ as a function of $p$. 2) The value of $p$ that minimizes the previous area. 3) The geometric place of the $P$ points of intersection of the parallels drawn by $A'$ and $C'$, respectively on the sides $AB$ and $AC$, when $p$ is between 0 and 1.