⁦Constructing an angle and triangles with compasses and ruler⁩

⁦Construction problems are solved with compasses and an unmarked ruler only⁩. ⁦The ruler draws a line through one or two points; the compasses draw a circle with a given centre and a radius equal to a given segment⁩. ⁦Each problem has four steps⁩: ⁦analysis, construction, proof, investigation⁩.

⁦An angle equal to a given angle⁩. ⁦Given angle ABC and ray DK⁩. ⁦Draw a circle of radius r with centre B (it cuts the arms of the angle) and a circle of the same radius with centre D⁩. ⁦Call M where it cuts ray DK⁩. ⁦Then draw a circle with centre M and radius equal to the distance between the two points where the first circle cut the arms⁩. ⁦It cuts the second circle at E⁩. ⁦Angle MDE is the one⁩: ⁦the triangles are equal by three sides⁩.

⁦A triangle from two sides and the angle between them, or from a side and two adjacent angles⁩. ⁦First lay the angle (or the two sides), then mark the lengths on the sides with the compasses⁩. ⁦Each such triangle is unique up to congruence (the first and second congruence tests)⁩.

⁦A right triangle from the hypotenuse and a leg⁩. ⁦Draw two perpendicular lines m and n, meeting at C⁩. ⁦On m lay the leg CA = b⁩. ⁦Draw a circle with centre A and radius equal to the hypotenuse c⁩. ⁦It cuts n at B1 and B2⁩. ⁦Both triangles ACB1 and ACB2 work and are congruent, so there is one solution⁩.