Geometry

Constructions

The classic ruler-and-compass constructions, taught one movement at a time.

Watch each one, then draw it yourself with a real compass.

Step 1:
Pick

Choose a construction. Each card says how many steps it takes.

Step 2:
Follow

Every step is one movement of the compass, the ruler or the pencil.

Step 3:
Try it

Draw it on paper, or open it on the Geometry Board.
What it does.
Teaches each ruler-and-compass construction step by step, with the compass, ruler and pencil moving on a clean figure, and says why each step works.
When it helps.
When you want to draw an accurate figure by hand, or understand where a construction on the Geometry Board comes from.
What it won’t do.
It won't solve problems for you, and it never measures. Every construction uses only a straight edge and a compass.
Example.
A 15 degree angle: make 60 degrees, halve it, then halve it again.

All constructions

Basics

Angles

Lengths

Each starts from a given segment of length 1.

Shapes

Every Geometry Board construction

Each tool on the board with a Do it by hand card, as a full lesson on its own clean figure.

Points (4)

MidpointMakes the point exactly halfway between two points.11 steps1 parts
Divide in a ratioMakes the point that divides a segment in a ratio you type.25 steps5 parts
Harmonic conjugateMakes the harmonic conjugate of a point on the line through two others.18 steps7 parts
Centre of a circleMarks the centre of a circle.24 steps4 parts

Build (10)

Perpendicular lineDraws the line through a point at right angles to a line.12 steps1 parts
Parallel lineDraws the line through a point that runs alongside a line and never meets it.12 steps1 parts
Perpendicular bisectorDraws the line through the midpoint of two points, at right angles to the segment joining them.11 steps1 parts
Angle bisectorDraws the line that splits an angle into two equal angles.12 steps1 parts
External angle bisectorDraws the line that splits the outside angle at a vertex into two equal angles.25 steps3 parts
Foot of perpendicularMarks where the perpendicular from a point meets a line.13 steps2 parts
MedianDraws the line from a corner of a triangle through the midpoint of the opposite side.13 steps2 parts
AltitudeDraws the line from a corner of a triangle at right angles to the opposite side.12 steps1 parts
SymmedianDraws the median from a corner reflected in the angle bisector there.34 steps5 parts
Isogonal lineDraws the line through a vertex reflected in that angle's bisector.21 steps3 parts

Tangents and power (7)

Tangent at a pointDraws the tangent to a circle at a point on it.15 steps3 parts
Tangents from a pointDraws both tangents from a point outside a circle.19 steps4 parts
Common tangentsDraws every line that touches both of two circles.44 steps7 parts
Polar of a pointDraws the polar line of a point for a circle.17 steps3 parts
Pole of a lineMarks the pole of a line for a circle.31 steps6 parts
Radical axisDraws the line of points that have equal power for two circles.3 steps2 parts
Radical centreMarks the point that has equal power for three circles.47 steps5 parts

Circles (7)

Circle with centreDraws the circle with a centre you pick through a point you pick.3 steps1 parts
Circle through three pointsDraws the circle through three points.27 steps4 parts
Circle with copied radiusDraws a circle whose radius is copied from the distance between two points.4 steps3 parts
Circle on a diameterDraws the circle that has a segment as a diameter.14 steps2 parts
Arc with centreDraws an arc round a centre, counter-clockwise from a start point to a ray.5 steps3 parts
Arc through three pointsDraws the arc that runs from one point through a second to a third.26 steps4 parts
Circle of ApolloniusDraws the circle of points whose distances to two points are in a ratio you type.66 steps8 parts

Triangle centres (8)

CircumcentreMarks the circumcentre: the point the same distance from all three corners.23 steps3 parts
IncentreMarks the incentre: where the angle bisectors of a triangle meet.25 steps3 parts
CentroidMarks the centroid: where the medians of a triangle meet.27 steps4 parts
OrthocentreMarks the orthocentre: where the altitudes of a triangle meet.25 steps3 parts
IncircleDraws the circle inside a triangle that touches all three sides.42 steps6 parts
ExcircleDraws the circle outside a triangle that touches one side and the other two sides extended.47 steps8 parts
Nine-point circleDraws the circle through the midpoints of a triangle's sides and the feet of its altitudes.59 steps7 parts
Mixtilinear incircleDraws the circle that touches two sides at a corner and the circumcircle from inside.56 steps10 parts

More centres (11)

ExcentreMarks the centre of the excircle opposite a corner.31 steps5 parts
Nine-point centreMarks the centre of the nine-point circle.56 steps6 parts
Symmedian pointMarks the symmedian point, where the three symmedians meet.69 steps7 parts
Gergonne pointMarks where the lines from each corner to the incircle's touch point on the opposite side meet.56 steps10 parts
Nagel pointMarks where the lines from each corner to an excircle's touch point on the opposite side meet.88 steps12 parts
Fermat pointMarks the first isogonic centre, where the lines to the far corners of outward equilateral triangles meet.25 steps5 parts
Feuerbach pointMarks where the incircle touches the nine-point circle.100 steps13 parts
Isogonal conjugateMarks the isogonal conjugate of a point in a triangle.43 steps7 parts
Isotomic conjugateMarks the isotomic conjugate of a point in a triangle.43 steps7 parts
Miquel pointMarks the point shared by the circumcircles of the four triangles that four lines make.54 steps8 parts
Centre of spiral similarityMarks the centre of the turn and scaling that takes one segment onto another.58 steps8 parts

Shapes (9)

Equilateral triangle on a segmentBuilds an equilateral triangle on a segment, with its corners and sides.12 steps2 parts
Square on a segmentBuilds a square on a segment, with its corners and sides.40 steps6 parts
Regular polygon on a segment: 5 sidesBuilds a regular polygon on a segment, with its corners and sides.61 steps6 parts
Regular polygon on a segment: 6 sidesBuilds a regular polygon on a segment, with its corners and sides.32 steps4 parts
Regular polygon on a segment: 8 sidesBuilds a regular polygon on a segment, with its corners and sides.53 steps8 parts
Regular polygon on a segment: 10 sidesBuilds a regular polygon on a segment, with its corners and sides.82 steps9 parts
Regular polygon on a segment: 12 sidesBuilds a regular polygon on a segment, with its corners and sides.71 steps8 parts
Triangle from three segmentsBuilds a triangle whose sides are copies of three segments.22 steps5 parts
Complete a parallelogramBuilds the fourth corner of a parallelogram, with its sides.13 steps6 parts

Transform (8)

ReflectMakes the mirror image of a point, line, circle, arc or polygon.8 steps4 parts
Rotate by an angleTurns an object about a centre by an angle you type.14 steps4 parts
Rotate by an angle on the boardTurns an object about a centre by an angle already on the board.20 steps5 parts
TranslateSlides an object by the vector from one point to another.9 steps1 parts
DilateScales an object from a centre by a factor you type.27 steps6 parts
Dilate by a ratio of segmentsScales an object from a centre by the ratio of two segments.29 steps7 parts
Invert in a circleInverts a point, a full line or a full circle in a circle.20 steps4 parts
Spiral similarityTurns and scales an object by the spiral similarity that takes A to C and B to D.29 steps5 parts