the website sciencevsmagic is a most helpful resource: 40 construction tasks to attempt

Patrick Honner's blog contains the suggestion that students can create the angle bisectors (and so the incentre) by paper folding:

and the line bisectors

to construct the circumcentre

to practice the (bisector) constructions using compasses and a straight edge, students can be asked to construct all three angle bisectors of a triangle

with sufficient accuracy the lines will meet at a point, the 'incentre', and this circle can be drawn:

this probably needs a sheet of plain paper

possibly informing students that they can vary their compass radius for each side bisection

- inside the triangle
- outside it
- (borderline case) on one of the sides of a triangle

noting the special case where the lines meet on one of the sides of the triangle

students can also be asked to bisect two joined ('hinged') lines - to find the centre of a circle that passes through the three points A, B and C (i.e. the circumcircle):

explore how three points determine a circle

usually...

further tasks, practicing these skills, could be to:

- construct a regular hexagon
- construct a 30 degree angle (and a 45 degree angle)
- construct an equilateral triangle

- construct a parallel to a line through a given point

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