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Illustration of a circle with diameter BC, chord DF, secant MN, and tangent HK.

Chords, Secants, Diameters, and Tangents of a Circle

Illustration of a circle with diameter BC, chord DF, secant MN, and tangent HK.

Illustration of a circle with diameter AB. Radii are RO, SO, AO, and BO. ED is a chord.

Chords, Diameters, and Radii of a Circle

Illustration of a circle with diameter AB. Radii are RO, SO, AO, and BO. ED is a chord.

Illustration of a circle with central angle AOB.

Central Angle in a Circle

Illustration of a circle with central angle AOB.

This pocket watch has a unique design on its back of leaves and circles. The circular handle at the top of the watch allows it to be attached to a waistcoat, lapel or belt loop using a chain.

Pocket Watch

This pocket watch has a unique design on its back of leaves and circles. The circular handle at the…

Illustration of a circle with center O and diameters AB and CD perpendicular to each other.

Circle With 2 Perpendicular Diameters

Illustration of a circle with center O and diameters AB and CD perpendicular to each other.

Illustration used to show that "In equal circles, or in the same circle, if two chords are equal, they subtend equal arcs; conversely, if two arcs are equal, the chords that subtend them are equal."

Equal Chords in Equal Circles Theorem

Illustration used to show that "In equal circles, or in the same circle, if two chords are equal, they…

Illustration used to show that "In equal circles, or in the same circle, if two chords are unequal, the greater chord subtends the greater minor arc; conversely, if two minor arcs are unequal, the chord that subtends the greater arc is the greater."

Unequal Chords in Circles Theorem

Illustration used to show that "In equal circles, or in the same circle, if two chords are unequal,…

Illustration used to show that "In equal circles, or in the same circle, if two chords are equal, they are equally distant from the center; conversely, if two chords are equally distant from the center, they are equal."

Equal Chords in Equal Circles Theorem

Illustration used to show that "In equal circles, or in the same circle, if two chords are equal, they…

Illustration used to show that "In equal circles, or in the same circle, if two chords are unequal, the greater chord is at the less distance from the center."

Unequal Chords in Equal Circles Theorem

Illustration used to show that "In equal circles, or in the same circle, if two chords are unequal,…

Illustration used to show that "In equal circles, or in the same circle, if two chords are unequal, the greater chord is at the less distance from the center."

Unequal Chords in Equal Circles Theorem

Illustration used to show that "In equal circles, or in the same circle, if two chords are unequal,…

Illustration of a quadrilateral circumscribed about a circle. This could also be described as a circle inscribed in a quadrilateral.

Quadrilateral Circumscribed About a Circle

Illustration of a quadrilateral circumscribed about a circle. This could also be described as a circle…

Illustration used to show that "If two tangents are drawn from any given point to a circle, those tangents are equal."

Equal Tangents to Circle Theorem

Illustration used to show that "If two tangents are drawn from any given point to a circle, those tangents…

Illustration used to inscribe a circle in a given triangle.

Construction Used to Inscribe a Circle in a Triangle

Illustration used to inscribe a circle in a given triangle.

Illustration used to escribe a circle to a given triangle. "A circle which is tangent to one side of a triangle and to the the other two sides prolonged is said to be escribed to the triangle."

Construction Used to Escribe a Circle to a Triangle

Illustration used to escribe a circle to a given triangle. "A circle which is tangent to one side of…

Illustration used to circumscribe a circle about a given triangle.

Construction Used to Circumscribe a Circle About a Triangle

Illustration used to circumscribe a circle about a given triangle.

Illustration used to prove "If two circumferences meet at a point which is not on their line of centers, they also meet in one other point."

Circumferences of 2 Circles

Illustration used to prove "If two circumferences meet at a point which is not on their line of centers,…

Illustration of two circles that are externally tangent to each other.

2 Externally Tangent Circles

Illustration of two circles that are externally tangent to each other.

Illustration of two circles that are internally tangent to each other.

2 Internally Tangent Circles

Illustration of two circles that are internally tangent to each other.

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured by its intercepted arc."

Central Angles and Arcs in a Circle

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured…

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured by its intercepted arc." Angle AOB and angle COE are commensurable.

Central Angles and Arcs in a Circle

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured…

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured by its intercepted arc." Angle AOB and angle COE are incommensurable.

Central Angles and Arcs in a Circle

Illustration of a circle that can be used to show that an "angle at the center of a circle is measured…

Illustration of a circle with an inscribed angle that can be used to prove that "An inscribed angle is measured by one half of its intercepted arc." In this case, one side of angle ABC passes through the center of the circle.

Inscribed Angle in a Circle Proof

Illustration of a circle with an inscribed angle that can be used to prove that "An inscribed angle…

Illustration of a circle with an inscribed angle that can be used to prove that "An inscribed angle is measured by one half of its intercepted arc." In this case, center O lies within angle ABC.

Inscribed Angle in a Circle Proof

Illustration of a circle with an inscribed angle that can be used to prove that "An inscribed angle…

Illustration of a circle with an inscribed angle that can be used to prove that "An inscribed angle is measured by one half of its intercepted arc." In this case, center O lies outside angle ABC.

Inscribed Angle in a Circle Proof

Illustration of a circle with an inscribed angle that can be used to prove that "An inscribed angle…

Illustration of a circle used to prove "All angles inscribed in the same segment are equal."

Angles Inscribed in the Same Segment Circle Proof

Illustration of a circle used to prove "All angles inscribed in the same segment are equal."

Illustration of a circle used to prove "Any angle inscribed in a semicircle is a right angle."

Right Angles Inscribed in Semicircle Proof

Illustration of a circle used to prove "Any angle inscribed in a semicircle is a right angle."

Illustration of a circle used to prove "Any angle inscribed in a segment less than a semicircle is an obtuse angle."

Obtuse Angles Inscribed in Circle Proof

Illustration of a circle used to prove "Any angle inscribed in a segment less than a semicircle is an…

Illustration used to show how "to construct a tangent to a circle from a point outside."

Construction of a Tangent to a Circle

Illustration used to show how "to construct a tangent to a circle from a point outside."

Illustration used to show how "to construct a common external tangent to two given circles."

Construction of an External Tangent to 2 Circles

Illustration used to show how "to construct a common external tangent to two given circles."

Illustration used to show how to construct two common external tangents to two given circles.

Construction of an 2 External Tangents to 2 Circles

Illustration used to show how to construct two common external tangents to two given circles.

"Byzantine Capital. The leading forms of the Byzantine style are the round arch, the circle, and in particular the dome." -Vaughan, 1906

Byzantine Capital

"Byzantine Capital. The leading forms of the Byzantine style are the round arch, the circle, and in…

"The first change in the parent-cell is that by which it becomes broken up into a mass of cells, each of which is just like itself. This process is called segmentation of the vitellus; each one of the numerous resulting cells is called a cleavage-cell. The nucleus of the parent-cell divides into two; each attracts its half of the yelk; the halves furrow apart and there are now two cleavage cells in place of the one parent-cell a furrow at right angles to the first, and redivision of the nuclei; results in four cleavage-cells. Radiating furrows intermediate to the first two bisect the four cells, and would render eight cells, were not these simultaneously doubled by a circular furrow which cleaves each, with the result of sixteen cleavage-cells. So the subdivision goes on until the parent-cell becomes a mass of cells. This particular kind of cleavage, by radiating and concentric furrowing, is called discoidal, and the resulting heap of little cells assumes the figure of a thin, flat, circular disc. Segmentation of the vitellus, in whatever manner it may go on, results in a mulberry-like mass of cleavage-cells; and the original cytula has become what is called a morula. This process is shown closely here." Elliot Coues, 1884

The Segmentation of the Vitellus

"The first change in the parent-cell is that by which it becomes broken up into a mass of cells, each…

This is a plan of the Salisbury Cathedral, England. It is an example of English Gothic architecture. The scale is in feet. "The square eastern termination, the less ambitious height, and the comparatively simple buttress–system, combine to give the English Gothic cathedral an air of great repose than is found in the magnificent triumphs of French Gothic art." The grouping "of 'lancet' windows, the piercing of the wall above them with the foiled circles, and the combination of the whole under an enclosed arch, soon led to the introduction of tracery, for which the design of earlier triforium arcades had also afforded a suggestion."

Plan of Salisbury Cathedral, 1075–1092

This is a plan of the Salisbury Cathedral, England. It is an example of English Gothic architecture.…

"Meroblastic ovum (yelk) of domestic fowl, bat. size, in section; after haeckel. a, the thin yelk-skin, enclosing the yellowfood-yelk, which is deposited in concentric layers, c, d; b, the cicatricle or tread with its nuclues, whence passes a cord of white yelk (here represented in black) to the central cavity, d'" Elliot Coues, 1884

Fowl Ovum

"Meroblastic ovum (yelk) of domestic fowl, bat. size, in section; after haeckel. a, the thin yelk-skin,…

Oval with dotted vertical and horizontal lines that are lines of symmetry.

Lines of Symmetry, Oval With

Oval with dotted vertical and horizontal lines that are lines of symmetry.

A standard sized spring bow with bow points or spacers to draw circles.

Spring Bow with Bow Points

A standard sized spring bow with bow points or spacers to draw circles.

A standard sized pencil point spring bow to draw circles.

Pencil Point Spring Bow

A standard sized pencil point spring bow to draw circles.

A standard sized pen point spring bow to draw circles.

Pen Point Spring Bow

A standard sized pen point spring bow to draw circles.

A hook-spring pen point to draw circles.

Pen Point Hook-Spring Bow

A hook-spring pen point to draw circles.

A hook-spring with pencil point to draw circles.

Pencil Point Hook-Spring Bow

A hook-spring with pencil point to draw circles.

A hook-spring with bow point to draw circles.

Hook-Spring Bow with Bow Points

A hook-spring with bow point to draw circles.

A center screw bow to draw circles. The adjustable screw is in the middle of the bow.

Center Screw Bow

A center screw bow to draw circles. The adjustable screw is in the middle of the bow.

Compass beam is used for large circles a compass or a lengthening bar cannot draw.

Beam Compass

Compass beam is used for large circles a compass or a lengthening bar cannot draw.

Drop pen, or rivet pen, can make small circles faster than the bow pen. The pen is useful in bridge work, constructional work, and topographical drawing.

Drop Pen

Drop pen, or rivet pen, can make small circles faster than the bow pen. The pen is useful in bridge…

When drawing the circle, the compass is turned by the handle with the thumb and forefinger in a clockwise motion.

Drawing a Circle Using Compass

When drawing the circle, the compass is turned by the handle with the thumb and forefinger in a clockwise…

"Circles up to perhaps three inches in diameter may be drawn with the legs straight but for larger sizes both the needle-point and the pencil leg should be at the knuckle joints so as to be perpendicular to the paper." — French, 1911

Drawing Large Circles with Compass

"Circles up to perhaps three inches in diameter may be drawn with the legs straight but for larger sizes…

Lengthening bar is used to draw circles bigger than 10 inches.

Use of Lengthening Bar

Lengthening bar is used to draw circles bigger than 10 inches.

Bow instruments can be used for drawing small circles. Hold the the left hand and spin the nut in or out with the finger to avoid wear and stripping the thread. "Small adjustments should be made with one hand, with needle point in position on the paper." — French, 1911

Drawing Small Circles with Bow Instrument

Bow instruments can be used for drawing small circles. Hold the the left hand and spin the nut in or…

"Draw Horizontal line throuch center of space. On it mark off radii for six concentric circles 1/4" apart. In drawing concentric circles always draw the smallest first. The dotted circles are drawn in in pencil with long dashes, and inked as shown." —French, 1911

Drawing Concentric Circles with Compass

"Draw Horizontal line throuch center of space. On it mark off radii for six concentric circles 1/4"…

"On horizontal center line mark off eleven points 1/4" apart, beginning at left side of space. Draw horizontal limiting lines (in pencil only) 1 1/2" above and below center line." —French, 1911

Drawing Concentric Arcs with Compass

"On horizontal center line mark off eleven points 1/4" apart, beginning at left side of space. Draw…

"On horizontal center line mark off eight points 3/8" apart, beginning at right side of space." —French, 1911

Drawing Concentric Arcs with Compass and Lengthening Bar

"On horizontal center line mark off eight points 3/8" apart, beginning at right side of space." —French,…

"On base AB, 3 1/2" long construct an equilateral triangle, using the 60-degree triangle. Bisect the angles with the 30-degree angle, extending the bisectors to the opposite sides. With these middle points of the sides as centers and radius equal to 1/2 the side, draw arcs cutting the bisectors. These intersection will be centers for the inscribed circles. With centers on the intersection of these circles and the bisectors, round off the points of the triangle as shown." —French, 1911

Drawing Tangent Circles and Lines with Compass and Triangles

"On base AB, 3 1/2" long construct an equilateral triangle, using the 60-degree triangle. Bisect the…

"In ordinary work the usual way of rectifying an arc is to step around it with the dividers, in spaces small enough as practically to coincide with the arc, and to step off the same number on the right line." —French, 1911

Rectifying Arc Using Dividers

"In ordinary work the usual way of rectifying an arc is to step around it with the dividers, in spaces…

The cone is sliced by a circle in a plane perpendicular to the axis. This can be drawn without knowledge of equations from analytic geometry.

Conic Section Using Circle

The cone is sliced by a circle in a plane perpendicular to the axis. This can be drawn without knowledge…

Using C as a center, draw two circles with different diameters. The intersection of the diameter lines will determine the points on the curve.

Determining Points on Ellipse Using Circles

Using C as a center, draw two circles with different diameters. The intersection of the diameter lines…

"A circle may be conceived as a polygon of an infinite number of sides. Thus to draw the involute of a circle divide it into a convenient number of parts, draw tangents at these points, lay off on these tangents the rectified lengths of the arch from the point of tangency to the starting point, and connect the points by a smooth curve." —French, 1911

Involute of Circle

"A circle may be conceived as a polygon of an infinite number of sides. Thus to draw the involute of…

"Divide the circumference into a number of equal parts, drawing the radii and numbering the points. Divide the radius No. 1 into the same number of equal parts, numbering from the center. With C as center draw concentric arcs intersecting the radii of corresponding numbers, and draw a smooth curve through these intersections." —French, 1911

Draw Spiral of Archimedes

"Divide the circumference into a number of equal parts, drawing the radii and numbering the points.…

The illustration shows the cylinder rolled out in a tangent plane of the base to create a development of the solid.

Development of Cylinder

The illustration shows the cylinder rolled out in a tangent plane of the base to create a development…

An illustration of finding an intersection of a cone and cylinder by either cutting the vertex of the cone and parallel to the cylinder; or by cutting circles from the right cone perpendicular to the axes.

Intersection of Cylinder and Cone

An illustration of finding an intersection of a cone and cylinder by either cutting the vertex of the…

An illustration of finding an intersection of a cone and cylinder by either cutting the vertex of the cone and parallel to the cylinder; or by cutting circles from the right cone perpendicular to the axes.

Intersection of Cylinder and Cone

An illustration of finding an intersection of a cone and cylinder by either cutting the vertex of the…