Circles as a Second World

This flow follows the book's expanding study of circles from centers and chords to tangents arcs and cyclic figures.

increasingBearing stren­gth in each eventdecreasing

Finding the Center

Finding the Center Perpendicular Bisectors Two chords are bisected and crossed by perpendicular lines. Their intersection is shown to be the unique point from which the relevant radii are equal. Geometric Location The center emerges not from inspection but from the invariant relations of chords and right angles.

Chord Interior

Chord Interior The Chord A straight line joining two points of a circle lies wholly within the circular boundary. The proof compares the chord with radii and uses triangle inequalities. Boundary and Region The result clarifies how a circle contains not only its circumference but the entire interior figure.

Center Chord Relations

Center Chord Relations Perpendicularity A diameter bisecting a chord is perpendicular to that chord, and the converse also holds. Equal distances from the center therefore become equivalent to equal chords. Structural Symmetry The circle's apparent symmetry is translated into exact relationships between lines and distances.

Tangent Discovery

Tangent Discovery Constructing Contact A tangent can be drawn from a point on or outside a circle. The radius to the point of contact is proved perpendicular to the tangent. Uniqueness The argument also shows that the perpendicular through contact must pass through the center, making tangency a rigid relation.

Cyclic Consequences

Cyclic Consequences Angles and Arcs Angles at the center and circumference are related through the arcs they intercept. Angles in the same segment become equal, while opposite angles of an inscribed quadrilateral are supplementary. Metric Completion Intersecting chords and secants convert circular position into rectangle and square relations.

at the first proposition of Book III

at the second proposition of Book III

at the third and fourteenth propositions of Book III

at the seventeenth through nineteenth propositions of Book III

at the twentieth through thirty-sixth propositions of Book III