Straight-edge Constructions in Geometry

People may Solve the following list useful. It is a relatively long list of good theorems for problem-solving with students if they have a ruler, compass and protractor. Some are very difficult without a lot of paper.

Basic Straightedge-and-compass Construction Theorems

• Through two points, Draw a unique line.
• With a centre and point/radius, construct a circle.
• Intersections of two lines, two circles, or a line and circle Determine constructible points.
• Construct an equilateral triangle on a given segment.
• Construct a perpendicular bisector.
• Construct the midpoint of a segment.
• Construct the centre of a circle Use perpendicular bisectors of chords.
• Construct a perpendicular from a point to a line.
• Construct a perpendicular to a line at a point on the line.
• Construct a parallel line through a given point.
• Construct an angle bisector.
• Copy a segment.
• Copy an angle.
• Copy a circle.
• Translate a segment.
• Use a collapsible compass versus a non-collapsing compass.

Locus and Equidistance Theorems

• Circle locus theorem: points at a fixed distance from a centre lie on a circle.
• Perpendicular bisector theorem: a point is on the perpendicular bisector of a segment if and only if it is equidistant from the segment’s endpoints.
• Angle bisector locus theorem: a point is on an angle bisector if and only if it is equidistant from the two sides of the angle.
• Line equidistant from two points.
• Line equidistant from two lines.
• Point equidistant from a side of an angle and a point.
• Circle equidistant from several points.
• Circle tangent or equidistant to lines and circles.
• Midpoint as an equidistance construction.
• Reflection as equidistance from a mirror line.

Perpendicular, Parallel and Angle Theorems

• A straight angle is .
• Half of a straight angle is a right angle.
• Two lines perpendicular to the same line are parallel.
• Parallel lines preserve corresponding angles.
• Parallel lines preserve alternate angles.
• Perpendicular lines form right angles.
• Angle bisectors halve angles.
• Double-angle construction.
• Construct a angle.
• Construct a angle.
• Construct a angle.
• Construct a angle.
• Construct a angle.
• Construct a angle.
• Construct an angle.
• Construct a angle.
• Special constructible angle trisection.
• General angle trisection is impossible using only straightedge and compass.

Triangle Congruence and Similarity Theorems

• SSS triangle construction.
• SAS congruence.
• ASA congruence.
• RHS or HL congruence for right triangles.
• Corresponding parts of congruent triangles are congruent.
• Isosceles triangle theorem.
• Converse of the isosceles triangle theorem.
• Equilateral triangle properties.
• Triangle angle sum theorem.
• Exterior angle theorem.
• Similar triangles.
• Intercept theorem.
• Basic proportionality theorem.
• Triangle mid-segment theorem.
• Medial triangle theorem.
• Triangle by midpoints.
• Triangle by altitude feet.
• Triangle by tangent points.
• Triangle by excentres.
• Triangle by side and centroid.
• Right triangle construction from hypotenuse and leg.
• Right triangle construction from hypotenuse and altitude.

Right-triangle and Metric Theorems

• Pythagorean theorem.
• Converse of the Pythagorean theorem.
• -- triangle ratios.
• -- triangle ratios.
• Geometric mean theorem in right triangles.
• Altitude-to-hypotenuse theorem.
• Hypotenuse-leg right-triangle congruence.
• Construction of .
• Construction of .
• Construction of geometric means of segments.
• Construction of harmonic means of segments.
• Construction of third proportionals.
• Construction of fourth proportionals.
• Construction of ratios such as .

Triangle Centres and Special Triangle Circles

• Circumcentre theorem: the perpendicular bisectors of a triangle meet at the circumcentre.
• Incentre theorem: the angle bisectors of a triangle meet at the incentre.
• Excentre theorem: internal and external angle bisectors meet at the excentres.
• Orthocentre theorem: the altitudes of a triangle meet at the orthocentre.
• Centroid theorem: the medians of a triangle meet at the centroid.
• The centroid divides each median in the ratio .
• Nine-point circle theorem.
• Incircle construction.
• Excircle construction.
• Circumscribed circle construction.
• Circumscribed triangle construction.
• Inscribed triangle construction.
• Circumscribed square construction.
• Inscribed square construction.
• Torricelli point construction.
• Fermat point construction.
• Triangle cleaver construction.

Circle Theorems

• Radius-to-tangent theorem: a tangent is perpendicular to the radius at the point of tangency.
• Converse tangent theorem.
• Tangents from the same external point are equal.
• Construct a tangent to a circle at a point.
• Construct a tangent from a point to a circle.
• Construct inner common tangents of two circles.
• Construct outer common tangents of two circles.
• Construct a circle tangent to a line.
• Construct a circle tangent to a line and a circle.
• Construct a circle tangent to two circles.
• Construct a circle tangent to three lines.
• Construct a circle through two points tangent to a line.
• Chord midpoint theorem: the perpendicular from the centre to a chord bisects the chord.
• Converse chord theorem: the perpendicular bisector of a chord passes through the circle centre.
• Equal chords subtend equal arcs.
• Equal chords subtend equal angles.
• Central angle and arc relation.
• Inscribed angle theorem.
• Thales’ theorem.
• Secant bisection.
• Secant relationship theorem.
• Power of a point theorem.
• Annulus theorem.
• Arbelos configuration theorem.
• Line-circle intersection construction.
• Circle with centre constrained to a line.
• Compass-only circle-centre construction.
• Napoleon’s problem.

Quadrilateral Theorems

• Parallelogram opposite sides are parallel.
• Parallelogram opposite sides are equal.
• Parallelogram diagonals bisect each other.
• A quadrilateral with one pair of opposite sides both parallel and equal is a parallelogram.
• Rectangle properties.
• Rhombus properties.
• Rhombus diagonals are perpendicular bisectors of each other.
• Square properties.
• Square construction from two adjacent vertices.
• Square construction from two opposite vertices.
• Square construction from midpoints of opposite sides.
• Square construction from midpoints of adjacent sides.
• Square construction from four points.
• Square construction inside a triangle.
• Square construction from an existing square.
• Kite construction.
• Lozenge construction.
• Trapezoid midline theorem.
• Midpoints of trapezoid bases theorem.
• The intersection of the legs of a trapezoid, the midpoints of the bases, and the intersection of the diagonals are collinear.
• Harmonic mean of trapezoid bases.
• Geometric mean of trapezoid bases.
• Varignon midpoint-parallelogram theorem.

Regular Polygon and Golden-ratio Constructions

• Regular hexagon construction from radius.
• Regular octagon construction.
• Octagon construction from square.
• Regular pentagon construction.
• Golden section construction.
• Golden ratio construction.
• Pentagon angle facts involving , and .
• Circumscribed regular polygon constructions.
• Inscribed regular polygon constructions.

Transformational Geometry Theorems

• Point reflection.
• Line reflection.
• Reflection preserves distance.
• Reflection preserves angle.
• Reflection creates perpendicular bisectors.
• Translation preserves length.
• Translation preserves direction.
• Rotation preserves distance.
• Rotation preserves angle.
• rotation construction.
• rotation construction.
• Centre of rotation construction.
• Symmetry of four lines.
• Reflection method for shortest-path problems.
• Restricted-tool reflection constructions.
• Restricted-tool midpoint constructions.
• Restricted-tool tangent constructions.
• Restricted-tool perpendicular constructions.

Optimisation and Shortest-path Theorems

• Triangle inequality.
• Reflection principle for minimum path problems.
• Reflection principle for minimum perimeter problems.
• Heron’s shortest-path theorem.
• Minimum-perimeter triangle in an angle.
• Minimum-perimeter triangle associated with a given triangle.
• Perimeter bisector construction.
• Farthest point from the sides of an angle.
• Symmetry and triangle inequality method for minimum perimeter constructions.

Projective and Advanced Named Theorems

• Pascal’s theorem.
• Nine-point circle theorem.
• Torricelli point theorem.
• Fermat point theorem.
• Arbelos theorem.
• Power of a point theorem.
• Inscribed angle theorem.
• Intercept theorem.
• Thales’ theorem.
• Pythagorean theorem.
• Perpendicular bisector theorem.
• Triangle inequality.
• Golden section theorem.
• Mascheroni compass-only construction theorem.
• Poncelet-Steiner restricted straightedge construction theorem.

Constructible-number and Proportion Theorems

• Segment doubling.
• Segment trisection.
• Chord trisection.
• Third proportional construction.
• Fourth proportional construction.
• Harmonic mean construction.
• Geometric mean construction.
• Golden ratio construction.
• Square-root construction.
• Ratio construction.
• Secant-based proportional construction.
• Similar-triangle proportional construction.

Named Theorem Checklist

• Perpendicular bisector theorem.
• Angle bisector locus theorem.
• Isosceles triangle theorem.
• Equilateral triangle theorem.
• SSS congruence.
• SAS congruence.
• ASA congruence.
• RHS or HL congruence.
• Corresponding parts of congruent triangles are congruent.
• Triangle angle sum theorem.
• Exterior angle theorem.
• Pythagorean theorem.
• Special right-triangle ratio theorems.
• Thales’ theorem.
• Inscribed angle theorem.
• Chord midpoint theorem.
• Tangent-radius theorem.
• Equal tangents theorem.
• Power of a point theorem.
• Intercept theorem.
• Basic proportionality theorem.
• Triangle mid-segment theorem.
• Median theorem.
• Centroid theorem.
• Orthocentre theorem.
• Circumcentre theorem.
• Incentre theorem.
• Excentre theorem.
• Nine-point circle theorem.
• Parallelogram diagonal theorem.
• Rhombus diagonal theorem.
• Trapezoid midpoint theorem.
• Varignon midpoint-parallelogram theorem.
• Triangle inequality.
• Reflection principle for shortest paths.
• Heron’s shortest-path theorem.
• Fermat-Torricelli point theorem.
• Pascal’s theorem.
• Golden section theorem.
• Regular polygon construction theorems.
• Mascheroni compass-only construction theorem.
• Poncelet-Steiner restricted straightedge construction theorem.
• Arbelos theorem.