Question 49
The triangle $ABC$ below is equilateral: its three sides have equal length and all its angles are $60^\circ$. Mr. Simas marks any point $H$ on $BC$, then draws through $H$ a segment parallel to $AC$ ending at $I$ on $AB$, and a segment parallel to $AB$ ending at $J$ on $AC$, as shown:
a) If $AB=1$, calculate the perimeter of $AIHJ$.
b) Mr. Simas continues as in the next figure, drawing $LO$ and $MP$ perpendicular to $BC$.
Let $d=IL$, $f=JM$ and $x=OP$. Prove that
$$x=\frac{1+d+f}{2}.$$
a) If $AB=1$, calculate the perimeter of $AIHJ$.
b) Mr. Simas continues as in the next figure, drawing $LO$ and $MP$ perpendicular to $BC$.
Let $d=IL$, $f=JM$ and $x=OP$. Prove that
$$x=\frac{1+d+f}{2}.$$