Construct a Geometric Figure

Produce ruler-and-compass construction for specified figure → doc every step w/ Euclidean justification + valid. result vs. original spec.

Use When

• Given specific geometric elements (points, segments, angles) + asked to construct figure
• Tasked w/ producing classical Euclidean construction (bisectors, perpendiculars, tangents)
• Valid. whether figure constructible w/ straightedge + compass alone
• Gen construction instructions for ed / docs
• Convert geometric spec → ordered seq of primitive ops

In

• Required: Desc of target figure (e.g., "equilateral triangle w/ side length AB")
• Required: Given elements (points, segments, circles, angles as starting data)
• Optional: Out format (narrative prose, numbered steps, pseudocode, SVG coords)
• Optional: Justification detail level (terse, standard, rigorous w/ theorem citations)
• Optional: Include impossibility analysis if not constructible

Do

Step 1: Identify Given Elements + Target Figure

Parse problem statement → extract:

1. Given elements -- list every point, segment, angle, circle, length provided.
2. Target figure -- state precisely what must be constructed.
3. Constraints -- note additional conditions (congruence, parallelism, tangency, collinearity).

Express problem in standard form:

Given: Points A, B; segment AB; circle C1 centered at A with radius r.
Construct: Equilateral triangle ABC with AB as one side.
Constraints: C must lie on the same side of AB as point P (if specified).

Valid. all ref'd elements well-defined + consistent.

→ Clear, unambiguous restatement of construction problem w/ every given element cataloged + target figure precisely described.

If err: Problem statement ambiguous → list possible interpretations + req clarification. Given elements contradictory (e.g., triangle w/ side lengths 1, 1, 5) → state contradiction + halt.

Step 2: Verify Constructibility

Determine whether target can be constructed w/ straightedge + compass alone.

1. Check algebraic constraints. Length constructible iff lies in field extension of rationals obtained by successive sqrt. Requires cube roots / transcendental ops → impossible.

2. Known impossible constructions:

   • Trisecting general angle
   • Doubling cube (constructing cube root of 2)
   • Squaring circle (constructing sqrt(pi))
   • Regular n-gon when n ≠ product of power of 2 + distinct Fermat primes

3. Known constructible ops:

   • Bisecting any angle / segment
   • Constructing perpendiculars + parallels
   • Transferring given length
   • Regular n-gons for n in {3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, ...}
   • Any length expressible using +, -, *, /, sqrt over given lengths

4. Doc verdict w/ justification.

Constructibility analysis:
- Target: equilateral triangle on segment AB
- Required operations: circle-circle intersection (two arcs of radius AB)
- Algebraic degree: 2 (quadratic extension)
- Verdict: CONSTRUCTIBLE

→ Definitive yes/no verdict on constructibility, w/ brief justification citing relevant algebraic / classical result.

If err: Constructibility uncertain → reduce problem to known constructible primitives. Provably non-constructible → doc impossibility proof + suggest closest constructible approximation or alt method (e.g., neusis construction, origami).

Step 3: Plan Construction Sequence

Decompose target figure → seq of primitive ops.

1. Identify primitives needed. Every ruler-and-compass reduces to these atomic ops:

   • Draw line through two points
   • Draw circle w/ given center + radius (center + point on circumference)
   • Mark intersection of two lines
   • Mark intersection(s) of line + circle
   • Mark intersection(s) of two circles

2. Order ops. Each op must ref only points already existing (given or prev constructed). Build dep graph:

Step 1: Draw circle C1 centered at A through B.       [uses: A, B]
Step 2: Draw circle C2 centered at B through A.       [uses: A, B]
Step 3: Mark intersections of C1 and C2 as P, Q.      [uses: C1, C2]
Step 4: Draw line through P and Q.                     [uses: P, Q]

3. Minimize step count. Look for opportunities to combine ops or reuse prev constructed elements.

4. Annotate each step w/ geometric purpose (e.g., "This constructs perpendicular bisector of AB").

→ Ordered list of primitive ops where each step depends only on prev est'd elements, covering all parts of target.

If err: Decomp stalls → ID which part of figure can't be reached from current set of constructed points. Revisit Step 2 to confirm constructibility, or introduce auxiliary constructions (helper circles, midpoints, reflections) to bridge gap.

Step 4: Execute Construction Steps w/ Justification

Write each construction step in full, providing Euclidean justification.

Each primitive op → doc:

1. Op: what is drawn / marked.
2. Ins: which existing elements used.
3. Justification: which Euclidean proposition / theorem / property guarantees op produces claimed result.
4. Out: what new elements created.

Format each step consistent:

Step 3: Mark intersections of C1 and C2 as P and Q.
  - Operation: Circle-circle intersection
  - Inputs: C1 (center A, radius AB), C2 (center B, radius BA)
  - Justification: Two circles with equal radii whose centers are separated
    by less than the sum of their radii intersect in exactly two points,
    symmetric about the line of centers (Euclid I.1).
  - Output: Points P and Q, where AP = BP = AB (equilateral property).

Continue until target fully constructed. Complex figures → group related steps into phases (e.g., "Phase 1: Construct auxiliary perpendicular bisector", "Phase 2: Locate incenter").

→ Complete seq of justified construction steps that, executed in order, produce target. Every new point, line, circle accounted for.

If err: Can't provide justification for step → step may be invalid. Valid. geometric claim independently. Common errs: assuming two circles intersect when they don't (check distance between centers vs. sum/diff of radii), or assuming point lies on line w/o proof.

Step 5: Verify Construction Meets Specification

Confirm constructed figure satisfies all original reqs.

1. Check each constraint from Step 1 vs. constructed figure:

   • Congruence: verify equal lengths / angles via construction.
   • Parallelism/perpendicularity: confirm via construction method (e.g., perpendicular bisector guarantees 90 deg).
   • Incidence: verify req'd points lie on req'd lines / circles.

2. Count degrees of freedom. Constructed figure should have exactly number of free params implied by spec. Extra dof → spec under-determined. None + construction fails → spec over-determined / contradictory.

3. Test w/ specific coords (optional but rec'd for complex constructions):

Verification with coordinates:
Let A = (0, 0), B = (1, 0).
C1: x^2 + y^2 = 1
C2: (x-1)^2 + y^2 = 1
Intersection: x = 1/2, y = sqrt(3)/2
Triangle ABC: sides AB = BC = CA = 1. VERIFIED.

4. Doc verification result w/ clear pass/fail each constraint.

→ Every constraint from original spec verified, construction confirmed correct. Coord check (when done) matches geometric argument.

If err: Constraint fails → trace back through construction → find erroneous step. Common causes: incorrect intersection choice (wrong branch of circle-line intersection), sign err in coord verification, or missing auxiliary construction.

Check

• Problem restated in standard Given/Construct/Constraints form
• Constructibility analysis w/ clear verdict + justification
• Every construction step uses only prev est'd elements
• Every step includes op, ins, justification, out
• Justification cites relevant geometric principle (Euclid, theorem name, property)
• Target fully constructed (no missing components)
• All original constraints verified vs. completed construction
• No step relies on measurement, approximation, or non-constructible ops
• Step count reasonable for figure complexity

Traps

• Assume intersection exists: Two circles only intersect if distance between centers between |r1 - r2| + r1 + r2. Always valid. this before marking intersection points. Forgetting → constructions work on paper but fail geometrically.

• Wrong intersection branch: Circle-circle + line-circle intersections yield two points. Construction must specify which to use (e.g., "intersection on same side of AB as point P"). Ambiguous intersection choices → two valid but diff figures.

• Conflate construction w/ measurement: Ruler-and-compass doesn't allow measuring lengths / angles. Can't "measure segment AB, mark off same length." Use compass to transfer radius by drawing circle centered at new point through old endpoint.

• Skip constructibility check: Attempting to trisect general angle or construct regular heptagon wastes effort. Always valid. constructibility before beginning construction seq.

• Over-complicated seqs: Many constructions have elegant short solutions. Construction exceeds 15 primitive steps for standard figure → look for simpler approach. Classic sources (Euclid, Hartshorne) often provide minimal constructions.

• Implicit auxiliary elements: Failing to doc helper constructions (e.g., "extend line AB to point D") makes seq impossible to follow. Every element used must be explicit constructed.

→

• solve-trigonometric-problem - trig analysis often motivates / verifies constructions
• prove-geometric-theorem - constructions frequently appear as steps in geometric proofs
• create-skill - follow when packaging new construction as reusable skill