How to determine the curve? 
In the figure above, segment $PQ$ is determined by two points: $P: (t,0)$ and $Q: (1,t)$, where $t\in [0,1]$ continuously increases and decreases between $0$ and $1$.
Then this gives a close region swept by $PQ$, the upper edge of which is a curve.
How to determine the equation of the curve (maybe implicit form)?
My own method
Suppose the curve is $y=y(x)$, then for any fixed $x_0\in (0,1)$, there is a vertical line $x=x_0$, which intersects a bundle of such $PQ(t)$ segments:
$$y=\frac{t(x-t)}{1-t}$$
Easy to conclude that, the desired $y_0$ on the curve corresponding to $x_0\left(\in(0,1)\right)$ is the maximum of:
$$y_0= \max\limits_{t\in (0,1)}\frac{t(x_0-t)}{1-t}=2-2\sqrt{1-x_0}-x_0$$
How to use the envelope concept? Is there any elementary method since the area is $\frac{1}{6}$?

HINT
I would say
$F(x,y,t)=0\equiv t(x-t)-y(1-t)=0,\quad t\in\langle 0,1 \rangle$
$F'_t(x,y,t)=0\equiv x-2t+y=0$
Solving this system of equations for the x, y:
$x=t(2-t), y=t^2, \quad t\in\langle 0,1 \rangle$ is a parametric curve of equation to search.
Plot
