Constructing Parallels, Tangents, and Regular Figures

The Central Idea

A construction succeeds when each step creates a condition that forces the final figure to have the desired property.

In this lesson, the main conditions are:

• equal angles that force parallelism;
• perpendicular radius relationships that force tangency;
• equal radii and equal central angles that force regularity.

The guiding principle is:

Three Construction Families

We will study three closely related construction families:

1. constructing parallel lines;
2. constructing tangent lines;
3. constructing regular polygons and related regular figures.

Each combines classical compass-and-straightedge operations with theorems that guarantee correctness.

Why Parallel Construction Matters

Parallel lines preserve direction in Euclidean geometry.

They appear in:

• coordinate grids;
• parallelograms and trapezoids;
• similarity constructions;
• translations;
• projection methods;
• architectural and engineering layouts.

Parallel Lines through a Point

Given a line \(\ell\) and a point \(P\) not on \(\ell\), Euclidean geometry guarantees exactly one line through \(P\) parallel to \(\ell\).

The construction problem is to create that unique line using permitted operations.

Method 1: Double Perpendicular

One reliable way to construct a parallel is to use two perpendicular constructions.

1. Construct a line \(m\) through \(P\) perpendicular to \(\ell\).
2. Construct a line \(n\) through \(P\) perpendicular to \(m\).

Then

Why the Double-Perpendicular Method Works

In a Euclidean plane, if two lines are perpendicular to the same line, they are parallel.

Since

and

line \(n\) has the same direction as \(\ell\).

Method 2: Copy a Corresponding Angle

Parallelism can also be created by constructing equal corresponding angles.

1. Choose a point \(A\) on \(\ell\).
2. Draw transversal \(\overleftrightarrow{AP}\).
3. Copy at \(P\) the angle formed by \(\ell\) and the transversal at \(A\).
4. Draw the new line through \(P\) forming that copied angle.

Why Copying a Corresponding Angle Works

If a transversal cuts two lines and corresponding angles are congruent, then the two lines are parallel.

Thus the copied angle creates the condition needed for the converse of the Corresponding Angles Theorem.

Method 3: Copy an Alternate Interior Angle

A similar construction can use alternate interior angles.

Draw a transversal from the given line to the external point, then copy the appropriate interior angle on the opposite side of the transversal.

The resulting congruent alternate interior angles establish parallelism.

Parallel Construction Is Theorem-Driven

The new line is not parallel because it looks parallel.

It is parallel because the construction creates a condition such as

in a configuration where a parallel-line converse theorem applies.

Uniqueness of the Parallel

Through a point \(P\) outside a line \(\ell\), Euclidean geometry permits exactly one line parallel to \(\ell\).

Therefore different correct construction methods must produce the same geometric line.

Parallel Construction and Translation

A translation moves every point the same distance in the same direction.

Under translation, a line is mapped either to itself or to a parallel line.

This gives a transformation-based interpretation of parallelism.

Constructing a Parallelogram

Given three noncollinear points \(A\), \(B\), and \(D\), we can construct a parallelogram \(ABCD\).

1. Through \(B\), construct a line parallel to \(\overline{AD}\).
2. Through \(D\), construct a line parallel to \(\overline{AB}\).
3. Let the two constructed lines meet at \(C\).

Then \(ABCD\) is a parallelogram.

Why the Parallelogram Construction Works

By construction,

and

A quadrilateral with both pairs of opposite sides parallel is a parallelogram.

Constructing a Rectangle

Given one side \(\overline{AB}\):

1. Construct perpendiculars to \(\overline{AB}\) at \(A\) and \(B\).
2. Choose equal heights \(AD=BC\) along the perpendiculars.
3. Connect \(C\) and \(D\).

The resulting quadrilateral is a rectangle.

Why the Rectangle Construction Works

The two side lines through \(A\) and \(B\) are both perpendicular to \(\overline{AB}\), so they are parallel.

The equal copied heights place \(C\) and \(D\) at corresponding positions.

The top segment is parallel to the base, and all four angles are right angles.

What Is a Tangent?

A line is tangent to a circle if it intersects the circle at exactly one point.

That point is called the point of tangency.

The Radius–Tangent Theorem

If line \(\ell\) is tangent to a circle at point \(T\), and \(O\) is the center, then

The radius to the point of tangency is perpendicular to the tangent line.

The Converse Radius–Tangent Theorem

If \(T\) lies on a circle centered at \(O\), and a line through \(T\) is perpendicular to \(\overline{OT}\), then that line is tangent to the circle at \(T\).

This converse gives a direct construction method.

Constructing a Tangent at a Known Point on a Circle

Suppose \(T\) is a known point on a circle with center \(O\).

1. Draw radius \(\overline{OT}\).
2. Construct a line through \(T\) perpendicular to \(\overline{OT}\).

The constructed line is tangent to the circle at \(T\).

Why the Tangent-at-a-Point Construction Works

The construction gives

Since \(T\) lies on the circle and \(\overline{OT}\) is a radius, the converse radius–tangent theorem establishes that \(\ell\) is tangent at \(T\).

Distance from the Center to a Tangent

The shortest distance from a circle's center to a tangent line is the radius drawn perpendicular to the tangent.

Thus if the radius is \(r\), the perpendicular distance from the center to the tangent line is

Constructing Tangents from an External Point

Let \(P\) be outside a circle centered at \(O\).

We seek tangent points \(T\) satisfying

The construction uses a circle with diameter \(\overline{OP}\).

External Tangent Construction

1. Draw segment \(\overline{OP}\).
2. Construct its midpoint \(M\).
3. Draw the circle centered at \(M\) with radius \(MO=MP\).
4. Let this auxiliary circle intersect the original circle at \(T_1\) and \(T_2\).
5. Draw \(\overline{PT_1}\) and \(\overline{PT_2}\).

These are the two tangent segments from \(P\).

Why the External Tangent Construction Works

Because \(T_1\) lies on the circle with diameter \(\overline{OP}\), \(\angle OT_1P\) is a right angle by Thales' Theorem.

Therefore,

Since \(T_1\) lies on the original circle, \(PT_1\) is tangent.

The Second Tangent

The same argument applies to \(T_2\):

Therefore \(PT_2\) is also tangent.

A point outside a circle generally determines two tangent lines.

Equal Tangent Segments

From the same external point \(P\), tangent segments to a circle have equal lengths.

Thus

Why the Tangent Segments Are Equal

The right triangles \(\triangle OPT_1\) and \(\triangle OPT_2\) share hypotenuse \(\overline{OP}\).

Also,

because both are radii.

The triangles are congruent, giving

When No Real Tangent from a Point Exists

If point \(P\) lies inside a circle, no real line through \(P\) can be tangent to that circle.

Every line through an interior point intersects the circle in two directions.

When Exactly One Tangent through a Point Exists

If \(P\) lies on the circle, exactly one tangent line passes through \(P\).

It is the line perpendicular to the radius through \(P\).

When Two Tangents through a Point Exist

If \(P\) lies outside the circle, exactly two tangent lines pass through \(P\).

The external tangent construction identifies both.

Tangency as a Distance Condition

A line \(\ell\) is tangent to a circle of radius \(r\) centered at \(O\) if the perpendicular distance from \(O\) to \(\ell\) equals

Thus tangency can be expressed through a locus-and-distance condition.

Coordinate Test for Tangency

For a circle centered at \((h,k)\) with radius \(r\), a line is tangent when the perpendicular distance from \((h,k)\) to the line is exactly \(r\).

This provides an algebraic counterpart to the geometric construction.

Circle and Tangent at a Coordinate Point

Suppose the circle is

and \(T=(3,4)\).

The radius \(\overline{OT}\) has slope

Therefore the tangent slope is

Tangent Equation at \((3,4)\)

Using point-slope form:

This line is perpendicular to the radius at the point on the circle, so it is tangent.

Common Tangents to Two Circles

Two circles can have several common tangent configurations depending on their positions and radii.

Possible common tangents include:

• external common tangents;
• internal common tangents.

The number of real common tangents changes as the circles move relative to each other.

What Is a Regular Polygon?

A regular polygon is both equilateral and equiangular.

Therefore:

and

Regular Figures and Circles

Regular polygons are naturally connected to circles.

Their vertices can be placed equally around a circumcircle, so equal central angles determine equal chords.

This makes the circle a natural construction framework.

Central Angle of a Regular \(n\)-gon

A full rotation has measure

Dividing it equally among \(n\) vertices gives central angle

Equal Central Angles Give Equal Chords

In the same circle, congruent central angles intercept congruent chords.

Therefore dividing a circle into \(n\) equal central angles creates \(n\) equal side lengths around the circumference.

Regularity from Rotational Symmetry

A regular \(n\)-gon is preserved by rotation through

about its center.

This rotational symmetry explains why corresponding sides and angles are equal.

Constructing an Equilateral Triangle in a Circle

A regular triangle has three equal central sectors.

Its central angle is

Connecting three equally spaced points on a circle produces an equilateral triangle.

Direct Equilateral Triangle Construction

Given a side \(\overline{AB}\):

1. Draw a circle centered at \(A\) with radius \(AB\).
2. Draw a circle centered at \(B\) with radius \(AB\).
3. Let an intersection be \(C\).
4. Connect \(A\), \(B\), and \(C\).

Then

Constructing a Square

Given side \(\overline{AB}\):

1. Construct perpendiculars to \(\overline{AB}\) at \(A\) and \(B\).
2. Copy the length \(AB\) along each perpendicular to locate \(D\) and \(C\).
3. Connect \(C\) and \(D\).

The resulting quadrilateral is a square.

Why the Square Construction Works

By construction,

and adjacent sides meet at right angles.

Thus the quadrilateral is both a rhombus and a rectangle, hence a square.

Square from a Circle

Another construction begins with a circle.

1. Construct one diameter.
2. Construct a perpendicular diameter through the center.
3. Connect the four endpoints consecutively.

The four central angles are each

The resulting inscribed quadrilateral is a square.

Constructing a Regular Hexagon

A regular hexagon has central angle

A special circle property makes this construction especially simple.

Hexagon Construction by Stepping the Radius

1. Draw a circle of radius \(r\).
2. Choose a point \(A\) on the circle.
3. Set the compass to the circle's radius \(r\).
4. Step that distance around the circumference six times.
5. Connect consecutive points.

The resulting polygon is a regular hexagon.

Why the Radius Steps a Regular Hexagon

In a circle, a chord equal to the radius subtends a central angle of

Six such angles complete the full rotation:

Thus six radius-length chords fit exactly around the circle.

Hexagon and Equilateral Triangles

Connecting the center of a regular hexagon to its six vertices divides the figure into six equilateral triangles.

Each has side length equal to the circumradius.

Constructing a Regular Dodecagon

Once a regular hexagon is inscribed, bisect each \(60^\circ\) central angle.

The new central angles have measure

Connecting the twelve equally spaced points gives a regular dodecagon.

Constructing an Octagon from a Square

Begin with a square inscribed in a circle.

The central angle between consecutive square vertices is

Bisect each central angle to create angles of

The eight resulting points form a regular octagon.

Repeated Angle Bisection Builds More Regular Polygons

If a regular \(n\)-gon is constructible, bisecting every central angle produces a regular

-gon.

This gives sequences such as

and

Constructing a Regular Pentagon

The regular pentagon is more subtle because its central angle is

Its geometry is closely connected to the golden ratio.

The Golden Ratio in a Pentagon

In a regular pentagon, a diagonal and a side satisfy

where

Because \(\sqrt5\) is constructible, the regular pentagon is classically constructible.

Regular Pentagon Construction Strategy

Classical constructions of the regular pentagon typically create the golden ratio from a chosen radius or diameter, then use the resulting chord length to step five equal vertices around a circle.

The exact sequence may vary, but the geometric objective is always the same:

From Pentagon to Decagon

Bisecting the central angles of a regular pentagon creates a regular decagon.

The decagon's central angle is

From Pentagon Families to More Polygons

Repeated central-angle bisection gives:

so constructibility propagates through powers of \(2\).

Not Every Regular Polygon Is Classically Constructible

A regular \(n\)-gon may be easy to draw approximately but impossible to construct exactly with only classical compass and straightedge.

Constructibility depends on the algebraic structure of

Gauss and the Regular \(17\)-gon

One famous result is that the regular \(17\)-gon is classically constructible.

This was a major discovery because \(17\) is not built merely from repeated doubling of the familiar \(3\)-, \(4\)-, or \(5\)-gon cases.

Gauss–Wantzel Criterion

A regular \(n\)-gon is classically constructible exactly when

where the \(p_i\) are distinct Fermat primes.

This is an advanced result linking Euclidean construction to algebra and number theory.

Examples of Constructible Regular Polygons

Classical examples include regular polygons with

sides, among many others.

A Regular \(7\)-gon Is Different

A regular heptagon has

Since \(7\) does not satisfy the classical constructibility criterion, an exact regular heptagon cannot be constructed with compass and straightedge alone.

Approximation versus Classical Construction

A regular heptagon can still be drawn to excellent visual accuracy using numerical or digital methods.

The impossibility statement concerns exact classical construction, not practical approximation.

Constructing a Regular Polygon from Its Center

If the center and circumradius are given, the task reduces to dividing the full angle around the center into equal parts.

Each central sector must have measure

Constructing a Regular Polygon from One Side

If one side length is given, the problem is different.

The construction must locate a center at the correct distance from the endpoints so that the given side becomes the required chord.

This usually involves constructing the appropriate central angle or circumradius.

Chord Length and Central Angle

For a circle of radius \(R\), a chord subtending central angle \(\theta\) has length

For a regular \(n\)-gon,

Therefore,

Apothem of a Regular Polygon

The apothem is the perpendicular distance from the center to a side.

For circumradius \(R\),

The apothem is also the inradius of the regular polygon.

Regular Polygons Have a Circumcircle

Every regular polygon can be inscribed in a circle whose center is the polygon's center.

Every vertex lies the same distance \(R\) from that center.

Regular Polygons Have an Incircle

Every regular polygon also has a circle tangent to all of its sides.

Its center is the same central point, and its radius is the apothem.

Tangents and Regular Polygons

The sides of a regular polygon circumscribed about a circle are tangent to that circle.

Each radius to a point of tangency is perpendicular to the corresponding side.

Constructing a Circumscribed Square

Given a circle:

1. Construct two perpendicular diameters.
2. At each of the four diameter endpoints, construct the tangent line to the circle.
3. Let adjacent tangent lines intersect.

The four intersections form a square circumscribed about the circle.

Why the Circumscribed Square Is Regular

Adjacent radii are perpendicular, and each tangent is perpendicular to its corresponding radius.

The tangent lines therefore form four right angles with equal symmetry around the center.

The resulting quadrilateral is a square.

Inscribed versus Circumscribed Regular Polygons

An inscribed regular polygon has all vertices on a circle.

A circumscribed regular polygon has all sides tangent to a circle.

The same center organizes both constructions.

Symmetry Axes of Regular Polygons

A regular \(n\)-gon has

reflection symmetries.

Depending on whether \(n\) is odd or even, the symmetry axes pass through combinations of vertices and side midpoints.

Regular Figures and Perpendicular Bisectors

The perpendicular bisector of any chord of a circle passes through the circle's center.

Therefore perpendicular bisectors of regular-polygon sides help locate the polygon's circumcenter.

Regular Figures and Angle Bisectors

In a regular polygon, symmetry axes through vertices bisect the corresponding interior angles.

Thus angle bisectors also pass through the center.

One Center, Many Roles

In a regular polygon, the same central point acts as:

• circumcenter;
• incenter;
• center of rotational symmetry;
• intersection of many perpendicular bisectors;
• intersection of many angle bisectors.

Constructing a Regular Figure by Symmetry

Many regular constructions can be viewed as repeatedly applying a rotation about a fixed center.

A rotation through

maps each vertex to the next.

Construction and Locus Intersections

Parallel, tangent, and regular-figure constructions can all be interpreted through loci.

For example:

• a tangent point satisfies a circle condition and a right-angle condition;
• a polygon center may lie on perpendicular bisectors of sides;
• a parallel line through a point satisfies an angle or direction condition.

Construction and Existence

A construction problem should always be checked for existence.

Examples:

• a point inside a circle has no tangent through it;
• a point on a circle has one tangent;
• a point outside a circle has two tangents;
• some regular polygons are not classically constructible.

Construction and Uniqueness

Some problems have unique solutions, while others have several.

A parallel through an external point is unique in Euclidean geometry.

Tangents from an external point generally come in a pair.

A regular polygon may have multiple orientations while remaining congruent.

Orientation Does Not Change the Mathematics

A square need not sit with horizontal and vertical sides.

A tangent need not look horizontal.

A parallel line need not slope upward.

Construction depends on exact relationships, not page orientation.

Construction and Scale

Enlarging or shrinking the initial data does not change the construction logic.

Similar versions of the figure preserve angles and proportional structure.

Dynamic Geometry as a Verification Tool

Dynamic geometry software can simulate classical constructions.

Dragging the initial points while construction dependencies remain intact can help verify that parallelism, tangency, or regularity persists.

This is useful experimental evidence, though formal justification still comes from geometry.

A Reliable Workflow for Constructing Parallels

1. Identify the given line and external point.
2. Choose a theorem that can certify parallelism.
3. Construct the required perpendicular or angle relationship.
4. Draw the candidate line through the required point.
5. State explicitly why the chosen theorem proves parallelism.

A Reliable Workflow for Constructing Tangents

1. Identify the circle center and the proposed tangent point or external point.
2. Use the radius–tangent right-angle condition.
3. Construct the required perpendicular or diameter-circle configuration.
4. Draw the candidate tangent line.
5. Verify that the radius at the point of contact is perpendicular to the line.

A Reliable Workflow for Constructing Regular Figures

1. Identify the number of sides \(n\).
2. Determine the required central angle \(360^\circ/n\).
3. Decide whether that angle or corresponding chord is classically constructible.
4. Construct equally spaced vertices on a circle.
5. Connect consecutive vertices.
6. Use equal radii, equal chords, and symmetry to justify regularity.

Common Error: Drawing a Parallel by Eye

Two lines that appear to have the same direction are not necessarily parallel.

A valid construction should create a theorem-supported condition such as equal corresponding angles or double perpendicularity.

Common Error: Using the Wrong Angle Converse

When constructing a parallel by angle copying, the copied angle must occupy the correct corresponding or alternate-interior position.

Equal angles in unrelated positions do not automatically imply parallelism.

Common Error: Tangent by Visual Contact

A line that seems to touch a circle at one point may actually cross it.

Use the radius–tangent perpendicular condition to establish exact tangency.

Common Error: Forgetting the Circle Center

Most tangent constructions depend on the radius to the point of tangency.

If the center is not known, it may need to be constructed first.

Common Error: Assuming an External Point Has One Tangent

A point outside a circle generally determines two tangent lines.

A complete construction should recognize both unless the problem specifies one by additional conditions.

Common Error: Confusing Chords and Tangents

A chord has both endpoints on the circle.

A tangent intersects the circle at exactly one point.

These are different geometric objects with different defining conditions.

Common Error: Calling an Equilateral Polygon Regular

Equal side lengths alone do not make every polygon regular.

A regular polygon must be both equilateral and equiangular.

Common Error: Calling an Equiangular Polygon Regular

Equal angles alone are also insufficient in general.

Regularity requires both equal sides and equal angles.

Common Error: Stepping an Arbitrary Chord Around a Circle

An arbitrary chord may not fit an integer number of times around the circumference.

To create a regular \(n\)-gon, the chord must correspond to central angle

Common Error: Assuming Every Regular Polygon Is Constructible

Approximate drawing is always possible numerically, but exact classical construction is more restrictive.

The Gauss–Wantzel criterion determines which regular \(n\)-gons are constructible.

Common Error: Erasing Construction Marks Too Soon

Auxiliary circles and arcs may provide the equal-distance or right-angle relationships needed for the proof.

Keep enough of the construction visible to justify the result.

Example: Construct a Parallel through \(P\)

Given line \(\ell\) and external point \(P\):

1. construct \(m\) through \(P\) with \(m\perp\ell\);
2. construct \(n\) through \(P\) with \(n\perp m\).

Therefore,

Example: Tangent at a Known Point

Given circle \(O\) and point \(T\) on the circle:

1. draw \(\overline{OT}\);
2. construct line \(\ell\) through \(T\) perpendicular to \(\overline{OT}\).

Then

Example: Two Tangents from an External Point

Given external point \(P\) and circle center \(O\), construct the circle with diameter \(\overline{OP}\).

Let its intersections with the original circle be \(T_1\) and \(T_2\).

Then

and

are tangent segments.

Example: Regular Hexagon

Draw a circle of radius \(r\) and use the same compass opening to step six consecutive chords around the circle.

Each side has length

and each central angle is

Therefore the polygon is regular.

Example: Regular Octagon

Construct two perpendicular diameters to obtain an inscribed square.

Bisect each \(90^\circ\) central angle.

The resulting eight central angles are

Connecting the eight points gives a regular octagon.

Example: Circumscribed Square

Construct perpendicular diameters of a circle, then construct tangents at their four endpoints.

The tangent intersections form a square surrounding the circle.

The original circle becomes the incircle of the square.

Practice 1: Parallel by Double Perpendicular

Line \(m\) is perpendicular to \(\ell\), and line \(n\) is perpendicular to \(m\).

What relationship holds between \(n\) and \(\ell\)?

Practice 2: Parallel by Angle Copying

A transversal cuts lines \(a\) and \(b\), and a pair of corresponding angles are constructed congruent.

What can be concluded?

Practice 3: Tangent at a Point

Point \(T\) lies on a circle centered at \(O\). A line through \(T\) is constructed perpendicular to \(\overline{OT}\).

What is the relationship between the line and the circle?

Practice 4: Tangents from an External Point

How many tangent lines generally pass through a point outside a circle?

Practice 5: Equal Tangent Segments

From external point \(P\), tangents touch a circle at \(A\) and \(B\).

State the length relationship between the tangent segments.

Practice 6: Regular Polygon Central Angle

What is the central angle of a regular \(12\)-gon?

Practice 7: Regular Hexagon

A regular hexagon is inscribed in a circle of radius \(8\).

What is the side length of the hexagon?

Practice 8: Constructing an Octagon

An inscribed square has central angles of \(90^\circ\).

What construction operation converts the square's four sectors into the eight equal sectors needed for a regular octagon?

Practice 9: Regularity

A pentagon has five congruent sides but its angles are not all congruent.

Is the pentagon regular?

Practice 10: Classical Constructibility

Can every regular polygon be constructed exactly with classical compass and straightedge?

Explain briefly.

Practice Answers

Practice 1

Two lines perpendicular to the same line are parallel in the Euclidean plane. Therefore,

Practice 2

By the converse of the Corresponding Angles Theorem,

Practice 3

A line through a point on a circle perpendicular to the radius at that point is tangent to the circle.

Practice 4

An external point generally determines

tangent lines to a circle.

Practice 5

Tangent segments from the same external point are congruent:

Practice 6

The central angle is

Therefore,

Practice 7

In a regular hexagon inscribed in a circle, each side equals the circumradius.

Practice 8

Bisect each central angle:

Therefore the required operation is

Practice 9

No. A regular polygon must be both equilateral and equiangular.

Practice 10

No. Exact classical constructibility depends on the number of sides. The Gauss–Wantzel criterion characterizes which regular \(n\)-gons can be constructed using compass and straightedge alone.

What to Remember

Parallel lines, tangents, and regular figures can all be constructed by creating exact defining relationships.

• Parallel lines can be constructed through double perpendicularity or suitable copied angles.
• A tangent at a point on a circle is perpendicular to the radius through that point.
• An external point generally determines two tangent lines to a circle.
• Tangent segments from the same external point are congruent.
• A regular polygon is both equilateral and equiangular.
• Equal central angles in one circle produce equal chords.
• A regular \(n\)-gon has central angle \(360^\circ/n\).
• Repeated angle bisection generates many larger constructible regular polygons.
• Not every regular polygon is classically constructible.
• Construction should always include both the geometric steps and the reason they work.

The central habit is: identify the defining condition first, then choose construction steps that force that condition to hold.

Where We Go Next

The constructions in this lesson repeatedly relied on sets of points satisfying exact distance, angle, and incidence conditions.

The next lesson, Locus Reasoning, studies this idea directly. We will treat geometric conditions as sets of possible points, combine multiple loci, and use intersections to solve construction and spatial reasoning problems.