Geometric constructions

This construction divides a line segment into any given ratio (such as $2:3$) using a straightedge and compass. Mark equal steps along an angled ray and draw parallel lines to divide the segment.

This method constructs a length $x = \sqrt{ab}$ (the geometric mean) from lengths $a$ and $b$. Draw a semicircle on diameter $a+b$; the perpendicular length $x$ erected from their junction equals $\sqrt{ab}$.

This method constructs tangent lines from an external point $P$ to a circle centered at $O$. Draw a circle on diameter $OP$; connecting $P$ to the intersection points forms right angles ($90^\circ$), creating the tangents.