Ruler and compass question

Question

Provide the exact list of steps needed to draw, using ruler and compass, a line $M$ through a given point $A$ and parallel to a given line $L$ (given by two points $B$ and $C$ on it). Assume that $A$ is not on $L$.

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I am completely new to this. So what are we allowed to start off with? Do we start off with the line $L$ and points $B$ and $C$ on this line. Then we have a point $A$ that is not on this line. Then we need to use these starting points so that we get this line $M$?

I am still pretty unsure how to do this...

Answer 1

Draw the line perpendicular to $BC$, we'll call it $D$. Then draw the circle about $A$ that intersects $D$ twice, at points $E$ and $F$. You can find the midpoint of these two points, $G$. Drawing the line through $G$ and $A$ will give you a line perpendicular to $D$.

If you have three lines, $X$, $Y$, $Z$, and you know that $X$ and $Y$ are perpendicular to $Z$, can you say anything about how $X$ and $Y$ relate?

Answer 2

The simplest way I know to construct line $M$ is to find the point $D$ such that $ABCD$ is a parallelogram. This can be done by intersecting the circle centred at $A$ with radius $p = BC$ and the circle centred at $C$ with radius $q = AB$.