loci and regions

harking back to a Euclidean tradition

the tasks at sciencevsmagic are interesting and challenging
if you have IT availability in the classroom

a good powerpoint is Dan Walker's
I've slightly adjusted it but this is his (admirable) work

a powerpoint showing the constructions (click and leave but click between slides)

a loop to remind students of the loci they need to know (which could be played whilst they work: click and leave)

some questions

alternative construction for the angle bisector

these are past paper questions from the (then) KS3 SAT papers in England

bisector concurrencies

it's not too difficult to show that the perpendicular bisectors of the sides of a triangle meet at one point

the circumcentre

a similar proof can be developed for showing that the angle bisectors meet at one point

the incentre

angle bisectors and trisectors

loci cartoons

cartoon
idea from John Mason
ask students to discuss what is going on and create ' the story'

three points determine a circle

how can you construct the triangle for which the circumcircle for the original triangle

is the incircle?

grid loci

these resources consider the various loci constructions on a grid
without needing to give out compasses...

students can plot the loci freehand

(the circle loci sometimes have two answers but only one of these is an integer pair)

angle bisectors

an intention of this task is that students use their angle bisecting skills to find and then possibly prove a relationship between two angles

a ppt is here

find the incentre and darw the circle just touching the three sides:

constructions

initial compass practice can be helpful for some students

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:

also for the perpendicular bisectors of the sides of a triangle - creating the 'circumcentre'

this probably needs a sheet of plain paper
possibly informing students that they can vary their compass radius for each side bisection

sometimes this circumcentre lies:

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

students can explore these three cases
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

how to construct a parallel to a line (CD) through a given point (A) :