# Construction ruler and compass [logic]

Carl is given three distinct non-parallel lines $$\ell_1, \ell_2, \ell_3$$ and a circle $$\omega$$ in the plane. In addition to a normal straightedge, Carl has a special straightedge which, given a line $$\ell$$ and a point $$P$$, constructs a new line passing through $$P$$ parallel to $$\ell$$. (Carl does not have a compass.) Show that Carl can construct a triangle with circumcircle $$\omega$$ whose sides are parallel to $$\ell_1,\ell_2,\ell_3$$ in some order.

Attempt: To build a triangle with a circle, just have $$3$$ points in that circle, at each point draw a line tangent to the circle and parallel to one of the given lines.

Am I right?

• well, no. for one thing, the triangle must be inside the circle. – Will Jagy Jan 8 at 22:29
• @WillJagy Why not? – Meulu Elisson Jan 8 at 22:32
• First we need to construct the midpoint of the circle, then I would construct a triangle with sides parallel to the given lines, construct its circumcenter, and use parallels to make a similar triangle attached to the given circle. – Berci Jan 8 at 22:41