Synthetic Construction Proof

Prove geometric theorems using classical Euclidean methods: auxiliary constructions, congruence, similarity, and angle chasing.

The Key Insight

Add helpful lines or points to reveal hidden structure. The right auxiliary construction transforms a difficult problem into an obvious one.

Step 1: Analyze the Given Figure

• List all given information
• Identify what needs to be proved
• Look for potential congruent or similar triangles
• Identify any special points (midpoints, incenters, circumcenters)

Questions to Ask

•Are there equal segments or angles given?
•Is there symmetry in the figure?
•Are there parallel or perpendicular lines?
•What triangles exist or could be formed?

Step 2: Plan Auxiliary Constructions

Common auxiliary constructions:

Technique	When to Use
Extend a line	To create exterior angles or meet another line
Draw a parallel	To transfer angles, create similar triangles
Draw a perpendicular	To create right angles, use Pythagorean theorem
Connect points	To form triangles from scattered points
Draw a midpoint	To use midpoint theorem, create isoceles triangles
Circumscribe circle	When equal angles or cyclic quadrilaterals involved
Drop altitude	To decompose into right triangles
Draw angle bisector	To create equal angles, use angle bisector theorem

Key Insight

The auxiliary construction should connect what you know to what you need.

Step 3: Apply Congruence Criteria

Triangle Congruence (≅)

Criterion	Conditions
SSS	Three pairs of equal sides
SAS	Two sides and included angle equal
ASA	Two angles and included side equal
AAS	Two angles and non-included side equal
HL	Hypotenuse and leg (right triangles only)

Triangle Similarity (~)

Criterion	Conditions
AA	Two pairs of equal angles
SAS~	Two sides proportional, included angle equal
SSS~	All three sides proportional

From Congruence/Similarity

•Congruent triangles → corresponding parts equal (CPCTC)
•Similar triangles → corresponding sides proportional

Step 4: Execute the Proof

• State each construction clearly
• Justify each step with a theorem or given
• Clearly mark the congruence/similarity
• Extract the conclusion using CPCTC or proportions

Common Theorems to Cite

Angle Theorems:

•Vertical angles are equal
•Alternate interior angles (parallel lines) are equal
•Corresponding angles (parallel lines) are equal
•Angles in a triangle sum to 180°
•Exterior angle equals sum of remote interior angles

Line Theorems:

•Midpoint theorem: Line joining midpoints is parallel and half the third side
•Angle bisector theorem: Divides opposite side in ratio of adjacent sides
•Triangle inequality: Sum of any two sides > third side

Circle Theorems:

•Inscribed angle = half central angle
•Angles in same arc are equal
•Tangent perpendicular to radius

Step 5: Lean Formalization

lean

import Mathlib.Geometry.Euclidean.Triangle
import Mathlib.Geometry.Euclidean.Angle.Oriented.Basic

-- Example: Isoceles triangle has equal base angles
-- Using Euclidean geometry in Lean
theorem isoceles_base_angles_equal
    {V : Type*} [NormedAddCommGroup V] [InnerProductSpace ℝ V]
    (A B C : V)
    (h : dist A B = dist A C) :
    -- angle ABC = angle ACB
    InnerProductGeometry.angle (B - A) (B - C) =
    InnerProductGeometry.angle (C - A) (C - B) := by
  sorry

Useful Mathlib Concepts

•EuclideanGeometry for geometric proofs
•dist for distances
•angle for angles
•Congruence expressed via equal distances and angles

Example: Prove Base Angles of Isoceles Triangle are Equal

Given: Triangle ABC with AB = AC Prove: ∠ABC = ∠ACB

Proof:

Construction: Draw the angle bisector from A to BC, meeting BC at D.

In triangles ABD and ACD:

•AB = AC (given)
•AD = AD (common side)
•∠BAD = ∠CAD (AD is angle bisector)

By SAS, △ABD ≅ △ACD.

Therefore, ∠ABD = ∠ACD (CPCTC), i.e., ∠ABC = ∠ACB. ∎

Output Format

code

**Theorem:** [Statement]

**Given:** [List given information]

**To Prove:** [Goal]

**Construction:** [Auxiliary elements added]

**Proof:**
[Step-by-step reasoning]

1. [Statement] ... [Justification]
2. [Statement] ... [Justification]
...

∴ △XYZ ≅ △ABC by [criterion]

**Conclusion:** [Final statement with CPCTC or proportions]

**Lean Proof:**
[Formal verification]

Common Pitfalls

•Assuming what you're proving: Don't use the conclusion in your proof
•Wrong congruence criterion: SSA (two sides and non-included angle) is NOT valid in general
•Unclear construction: State exactly what you're constructing and why it exists
•Missing justification: Every step needs a theorem or given as support
•Diagram dependency: Your proof must work even without the picture

Advanced Techniques

Proof by Construction of Congruent Copy

To prove AB = CD, construct triangle on CD congruent to triangle containing AB.

Angle Chasing

Chain together angle relationships: ∠A = ∠B (alternate interior) = ∠C (vertical) = ∠D (corresponding)

Auxiliary Circles

•Circumcircle through three points
•Incircle tangent to all sides
•Excircle opposite a vertex

Spiral Similarity

For similar triangles with common vertex, track the spiral similarity transformation.