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 "The diameter perpendicular to a chord bisects the chord and also its subtended arc."

Diameter Perpendicular to a Chord in a Circle

Illustration used to show that "The diameter perpendicular to a chord bisects the chord and also its…

Illustration used to show how to bisect a given arc.

Bisecting an Arc

Illustration used to show how to bisect a given arc.

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 used to show that "A tangent to a circle is perpendicular to the radius drawn to the point of tangency."

Tangent to Perpendicular Radius Circle Theorem

Illustration used to show that "A tangent to a circle is perpendicular to the radius drawn to the point…

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.

According to Norse mythology, in Elfland, elves and fairies would dance in a circle in the moonlight. "Whenever the fairies danced at night, the grass grew greener. Any one going through the forest the next day could tell where the elves had been by the rings of green grass." -Klugh, 1909

Elves and Fairies Dancing

According to Norse mythology, in Elfland, elves and fairies would dance in a circle in the moonlight.…

"A quadrant, or one fourth of a circle. The oblique lines indicate various angles with the base. The heavy line indicates the inclination of the earth's axis as compared with the plane of its orbit, which is represented by the base." -Wiswell, 1913

Inclination of Earth's Axis

"A quadrant, or one fourth of a circle. The oblique lines indicate various angles with the base. The…

"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…

An illustration of three shafts of wheat with a circle surrounding it.

Wheat & Circle Doodad

An illustration of three shafts of wheat with a circle surrounding it.

"Pitangus derbianus. Derby Flycatcher. Under parts light wood-brown, with an olive tinge; wings and tail the same, but the feathers extensively bordered without and within with chestnut, forming a conspicuous continuous area on the wing-quills in the closed wing, and on most of the wing and tail-feathers more extensively than the brown portion of the inner webs. Below from the breast, including lining of wings, clear and continuous lemon-yellow. Whole chin and throat pure white, widening behind up under ear-coverts. Top and sides of head black, a circle of white from forehead over eyes to nape white, the enclosed black enclosing black a lemon and orange patch. Or, middle of crown yellow and orange, enclosed and partly concealed in black, this black enclosed in white, then the long and broad black bar on side of head, separating the white of side of crown from that of side of throat. The coronal feathers lengthened and erectile as in a king-bird, or more so; crown-patch of same character but more extensive. Bill and feet black; iris hazel Sexes alike." Elliot Coues, 1884

Derby Flycatcher

"Pitangus derbianus. Derby Flycatcher. Under parts light wood-brown, with an olive tinge; wings and…

"Simorhynchus cristatellus. Crested Auk. Snub-nosed Auk. A beautiful crest of 12-20 slender feathers springing from the forehead, curling over forward in arc of a circle to fall gracefully upon the bill; this helmet is blackish; at full length about 2 inches long; the feathers are not filamentous, but have well formed webs, and are bundled or impacted together, owning to the oblique divergence of the webs from the shaft, as in the genus Laphortyx. A slender series of white filamentous feathers over and behind each eye, drooping downwards and backwards." Elliot Coues, 1884

Crested Auk in Winter

"Simorhynchus cristatellus. Crested Auk. Snub-nosed Auk. A beautiful crest of 12-20 slender feathers…

"Muscles of a bird (accipiter nisus), after Carus, Tab. Anat. Comp., 1828, pl. 4.   a, pharynx; b, trachea; e, hyoid bone; d, ear; e, humerous; f, radius; g, ulna; h, radial finger; i, tibia; k, metatarsus; l, hind toe; m, inner toe; n, middle toe; o, outer toe. 1, biventer cervicis, with central tendon 1 a, and upper 1 b, and lower 1 c, belly. 2, complexus. 3, flexor capitis lateralis. 4, flexor longus capitis. 5, extensor magnus cervicis. 6, descendens cervicis. 7, 7, semispinales. 8, flexorsuperior capitis. 9, flexor inferior or longus capitis. 10, 10, intertransversales. 11, levator coccygis. 12, depressor coccygis. 13, cruro-coccygeus (ilio-coccygeus?). 14, pubo-coccygeus. 15 ischio-coccygeus. 16, lateralis quartus (quadratus coccygis, to tail-feathers). 17, obliquus externus abdominis. 18, cucullaris (trapezius). 19, serratus magnus. 20, pectoralis major. 21, a, b, latissimus dorsi. 22, deltoid. 23, suprascapular. 24, coraco-brachialis. 25, biceps brachii. 26, supinatpr longus. 27, anconeus longus (part of "triceps"). 28, anconeus brevis. 29, anconeus brevissimus. 30 a, 30 b, tensor patagii, carpal and radial parts. 31, tensor patagii posterior. 32, extensormetacarpi longus. 33, extensor metacarpi brevis. 34 a, flexor digitorum sublimis. 34 b, flexor digitorum profundus. 34 c, flexor metacarpi radialis. 36, flexor (meta-) carpi ulnaris. 37, glutaeus maximus. 38, adductor femoris primus. 39, sartorius. 40, latissimus femoris. 41, gracilis = ambiens: only its tendon in sight. 42, vastus; 43, iceps cruris. 44, semimembranosus. 46,46,47, gastrocnemius. 48 digastricus (chief opener of the mouth). 49, temporal. 50, long ligament. 51, cutaneous muscle of scalp. 52, masseter. 53, a muscle of the hyoid bone. 54, tibialis anticus. 55, tibialis posticus. 56, extensor hallucis. 57, flexor hallucis. 58, flaxor digitorum profundus or perforans, seen in various places: long and short head, and several tendons. 59, extensor longus digitorum, tendons seen in various places 60, abductor digiti interni. 61,61,61, flexores digitorum perforati. 62, peronaeus. 63, abductor minimi digiti. 64, abductor hallucis." Elliot Coues, 1884

Eurasian Sparrowhawk Muscles

"Muscles of a bird (accipiter nisus), after Carus, Tab. Anat. Comp., 1828, pl. 4. a, pharynx; b, trachea;…

A circle divided into quarters with one quarter shaded.

Fraction Pie Divided into Quarters

A circle divided into quarters with one quarter shaded.

A circle divided into fifths with one fifth shaded.

Fraction Pie Divided into Fifths

A circle divided into fifths with one fifth shaded.

A circle divided into sixths with one sixth shaded.

Fraction Pie Divided into Sixths

A circle divided into sixths with one sixth shaded.

A circle divided into sevenths with one seventh shaded.

Fraction Pie Divided into Sevenths

A circle divided into sevenths with one seventh shaded.

A circle divided into eighths with one eighth shaded.

Fraction Pie Divided into Eighths

A circle divided into eighths with one eighth shaded.

A circle divided into ninths with one ninth shaded.

Fraction Pie Divided into Ninths

A circle divided into ninths with one ninth shaded.

A circle divided into tenths with one tenth shaded.

Fraction Pie Divided into Tenths

A circle divided into tenths with one tenth shaded.

A circle divided into elevenths with one eleventh shaded.

Fraction Pie Divided into Elevenths

A circle divided into elevenths with one eleventh shaded.

A circle divided into twelfths with one twelfth shaded.

Fraction Pie Divided into Twelfths

A circle divided into twelfths with one twelfth shaded.

A circle divided into quarters with two quarters shaded.

Fraction Pie Divided into Quarters

A circle divided into quarters with two quarters shaded.

A circle divided into quarters with two quarters shaded.

Fraction Pie Divided into Quarters

A circle divided into quarters with two quarters shaded.

A circle divided into fifths with two fifths shaded.

Fraction Pie Divided into Fifths

A circle divided into fifths with two fifths shaded.

A circle divided into sixths with two sixths shaded.

Fraction Pie Divided into Sixths

A circle divided into sixths with two sixths shaded.

A circle divided into sixths with two sixths shaded.

Fraction Pie Divided into Sixths

A circle divided into sixths with two sixths shaded.

A circle divided into sevenths with two sevenths shaded.

Fraction Pie Divided into Sevenths

A circle divided into sevenths with two sevenths shaded.

A circle divided into eighths with two eighths shaded.

Fraction Pie Divided into Eighths

A circle divided into eighths with two eighths shaded.

A circle divided into ninths with two ninths shaded.

Fraction Pie Divided into Ninths

A circle divided into ninths with two ninths shaded.

A circle divided into tenths with two tenths shaded.

Fraction Pie Divided into Tenths

A circle divided into tenths with two tenths shaded.

A circle divided into elevenths with two elevenths shaded.

Fraction Pie Divided into Elevenths

A circle divided into elevenths with two elevenths shaded.

A circle divided into twelfths with two twelfths shaded.

Fraction Pie Divided into Twelfths

A circle divided into twelfths with two twelfths shaded.

A circle divided into quarters with three quarters shaded.

Fraction Pie Divided into Quarters

A circle divided into quarters with three quarters shaded.

A circle divided into fifths with three fifths shaded.

Fraction Pie Divided into Fifths

A circle divided into fifths with three fifths shaded.

A circle divided into sixths with three sixths shaded.

Fraction Pie Divided into Sixths

A circle divided into sixths with three sixths shaded.

A circle divided into sixths with three sixths shaded.

Fraction Pie Divided into Sixths

A circle divided into sixths with three sixths shaded.

A circle divided into sevenths with three sevenths shaded.

Fraction Pie Divided into Sevenths

A circle divided into sevenths with three sevenths shaded.