Proof-Based Geometry – Complete Reference Guide

Euclid's Postulates

The Five Postulates

P1: A straight line can be drawn between any two points

P2: A line segment can be extended indefinitely

P3: A circle can be drawn with any center and radius

P4: All right angles are equal to one another

P5: (Parallel Postulate) If a line crosses two lines making interior angles summing to less than two right angles, those lines meet on that side

Common Notions

Things equal to the same thing are equal

Equals added to equals are equal

Key: P5 is equivalent to "through a point not on a line, exactly one parallel exists"

Angle Relationships

Basic Angle Types

Vertical: Opposite angles when lines cross; $\angle 1 = \angle 3$

Supplementary: Sum to $180°$

Complementary: Sum to $90°$

Linear Pair: Adjacent + supplementary

Parallel Lines + Transversal

Corresponding: Equal (same position)

Alternate Interior: Equal

Alternate Exterior: Equal

Co-Interior (Same-Side): Supplementary

Converse: If these angle relationships hold, lines are parallel

Triangle Congruence

Congruence Criteria

SSS: Three sides equal

SAS: Two sides + included angle

ASA: Two angles + included side

AAS: Two angles + non-included side

HL: Hypotenuse-Leg (right triangles only)

CPCTC

Corresponding Parts of Congruent Triangles are Congruent

Use after proving congruence to deduce equal parts

SSA: Does NOT prove congruence (ambiguous case)!

Triangle Similarity

Similarity Criteria

AA: Two angles equal

SAS~: Two sides proportional + included angle equal

SSS~: All three sides proportional

Properties

Corresponding angles are equal

Corresponding sides are proportional

$$\frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k$$

Area Relationship

$$\frac{\text{Area}_1}{\text{Area}_2} = k^2$$

AA suffices because angle sum is always $180°$

Triangle Angle Theorems

Angle Sum Theorem

Sum of interior angles = $180°$

$$\angle A + \angle B + \angle C = 180°$$

Exterior Angle Theorem

Exterior angle = sum of remote interior angles

$$\angle ACD = \angle A + \angle B$$

Triangle Inequality

Sum of any two sides > third side

$|a-b| < c < a+b$

Angle-Side Relationships

Larger angle opposite longer side

Larger side opposite larger angle

Proof tip: Extend side to create parallel line for angle sum

Right Triangle Theorems

Pythagorean Theorem

$$a^2 + b^2 = c^2$$

$c$ = hypotenuse, $a,b$ = legs

Altitude-on-Hypotenuse

Creates 3 similar triangles

$$h^2 = xy$$ (geometric mean)

$$a^2 = xc, \quad b^2 = yc$$

$x, y$ = segments of hypotenuse, $h$ = altitude

Special Right Triangles

45-45-90: $1 : 1 : \sqrt{2}$

30-60-90: $1 : \sqrt{3} : 2$

Converse: If $a^2 + b^2 = c^2$, triangle is right

Triangle Centers

Centroid (G)

Intersection of medians

Divides each median in ratio $2:1$

$G = \frac{A + B + C}{3}$

Incenter (I)

Intersection of angle bisectors

Center of inscribed circle

Circumcenter (O)

Intersection of perpendicular bisectors

Center of circumscribed circle

Orthocenter (H)

Intersection of altitudes

Euler Line: $H$, $G$, $O$ are collinear with $HG = 2 \cdot GO$

Ceva's & Menelaus' Theorems

Ceva's Theorem

Cevians $AD$, $BE$, $CF$ concurrent iff:

$$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = 1$$

Menelaus' Theorem

Points $D$, $E$, $F$ on sides (or extensions) collinear iff:

$$\frac{AF}{FB} \cdot \frac{BD}{DC} \cdot \frac{CE}{EA} = -1$$

(Using signed ratios)

Applications

Prove concurrency of lines in triangle

Prove collinearity of points

Remember: Ceva = 1 (concurrent), Menelaus = -1 (collinear)

Quadrilateral Properties

Parallelogram

Opposite sides parallel and equal

Opposite angles equal

Diagonals bisect each other

Rectangle

All angles = $90°$

Diagonals equal length

Rhombus

All sides equal

Diagonals perpendicular

Square

Rectangle + Rhombus properties

Trapezoid

One pair parallel sides

Midsegment = $\frac{b_1 + b_2}{2}$

To prove: Show diagonals have required property

Cyclic Quadrilaterals

Definition

All four vertices lie on a circle

Key Properties

Opposite angles are supplementary

$$\angle A + \angle C = 180°$$

Exterior angle = opposite interior angle

Ptolemy's Theorem

$$AC \cdot BD = AB \cdot CD + BC \cdot AD$$

Product of diagonals = sum of products of opposite sides

Tests for Cyclic

Opposite angles supplementary

Equal angles subtending same arc

Power: Ptolemy generalizes to all quadrilaterals with inequality

Circle Theorems

Central & Inscribed Angles

Central angle = arc measure

Inscribed angle = $\frac{1}{2}$ arc

Inscribed angles on same arc are equal

Thales' Theorem

Angle inscribed in semicircle = $90°$

Arc-Angle Relationships

Interior angle: $\frac{1}{2}(\text{sum of arcs})$

Exterior angle: $\frac{1}{2}|\text{difference of arcs}|$

Tangent Theorems

Tangent ⟂ radius at point of contact

Tangents from external point equal

Key: Inscribed angle theorem is foundation of circle proofs

Power of a Point

Definition

Power of $P$ w.r.t. circle: $d^2 - r^2$

$d$ = distance from $P$ to center

Chord-Chord

If chords $AB$ and $CD$ intersect at $P$:

$$PA \cdot PB = PC \cdot PD$$

Secant-Secant

If secants from $P$ through circle:

$$PA \cdot PB = PC \cdot PD$$

Secant-Tangent

If tangent $PT$ and secant $PAB$:

$$PT^2 = PA \cdot PB$$

Note: All are manifestations of same power formula

Proof Strategies

Direct Proof

Start from given, deduce conclusion

Chain of logical implications

Proof by Contradiction

Assume opposite of what you want to prove

Derive a contradiction

Conclude original statement is true

Proof by Cases

Split into exhaustive cases

Prove each case separately

Construction Proofs

Draw auxiliary lines (parallels, perpendiculars)

Extend segments, add circles

Strategy: Look for congruent or similar triangles first

Two-Column Proof Format

Structure

Left column: Statements

Right column: Reasons

Common Reasons

Given

Definition of (midpoint, bisector, etc.)

Reflexive Property ($a = a$)

Substitution

Transitive Property

Vertical Angles Theorem

SSS, SAS, ASA, AAS, HL

CPCTC

Flow Proof

Arrows show logical flow

Boxes contain statements + reasons

Tip: Plan backward from what you need to prove

Transformations

Isometries (Distance-Preserving)

Translation: Slide by vector

Rotation: Turn about a point

Reflection: Flip over a line

Glide Reflection: Reflect + translate

Properties Preserved

Distance, angle measure

Parallelism, collinearity

Area (for isometries)

Dilations

Scale by factor $k$ from center

Preserves angles, not distances

Area scales by $k^2$

Key: Two figures congruent iff one is isometric image of other

Coordinate Geometry Proofs

Key Formulas

Distance: $d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}$

Midpoint: $M = (\frac{x_1+x_2}{2}, \frac{y_1+y_2}{2})$

Slope: $m = \frac{y_2-y_1}{x_2-x_1}$

Proof Techniques

Parallel lines: equal slopes

Perpendicular: slopes multiply to $-1$

Congruent segments: equal distances

Strategic Placement

Place one vertex at origin

Align one side with axis

Use variables for general proof

Example: Prove diagonals bisect: show midpoints equal

Area Formulas & Proofs

Triangle Areas

$A = \frac{1}{2}bh$ (base × height)

$A = \frac{1}{2}ab\sin C$ (SAS)

Heron: $A = \sqrt{s(s-a)(s-b)(s-c)}$

$A = rs$ (inradius × semiperimeter)

Parallelogram

$A = bh$ (shear doesn't change area)

Trapezoid

$A = \frac{1}{2}(b_1 + b_2)h$

Circle

$A = \pi r^2$

Sector: $A = \frac{\theta}{360°} \pi r^2$

Proof method: Decompose into triangles

Compass & Straightedge

Basic Constructions

Perpendicular bisector of segment

Angle bisector

Perpendicular through point on line

Perpendicular from point to line

Copy an angle

Triangle Constructions

Circumcircle (⟂ bisectors meet)

Incircle (angle bisectors meet)

Impossible Constructions

Trisecting arbitrary angle

Doubling the cube

Squaring the circle

These require algebraic methods beyond compass/straightedge

Common Proof Patterns

Prove Segments Equal

Show they're corresponding parts of ≅ △s

Use properties of parallelograms

Show both equal to same third segment

Prove Angles Equal

Corresponding angles (parallel lines)

Vertical angles

Inscribed angles on same arc

CPCTC

Prove Lines Parallel

Show corresponding angles equal

Show alternate interior angles equal

Both perpendicular to same line

Always: Draw diagram, mark known info, work both directions

Advanced Theorems

Stewart's Theorem

For cevian $AD$ of length $d$:

$$b^2 m + c^2 n - a(d^2 + mn) = 0$$

Mass Point Geometry

Assign masses to vertices

Centroid at center of mass

Ratios from mass ratios

Nine-Point Circle

Passes through: midpoints of sides, feet of altitudes, midpoints of segments from vertices to orthocenter

Radius = $\frac{R}{2}$ (R = circumradius)

Simson Line

Feet of perpendiculars from point on circumcircle to sides are collinear

Competition: These appear in math olympiads

PROOF-BASED GEOMETRY Complete Reference