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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.

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 "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 three isosceles triangles with the same base AC but varying heights (as B gets farther from the base).

3 Isosceles Triangles Wit Equal Bases

Illustration of three isosceles triangles with the same base AC but varying heights (as B gets farther…

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.

Silhouette outlines of shapes (quadrilateral, hexagon, arrow, pentagon) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Shapes Silhouette Tangram Card #2

Silhouette outlines of shapes (quadrilateral, hexagon, arrow, pentagon) made from tangram pieces. Tangrams,…

Solutions for silhouette outlines of animals (bear, kangaroo, camel, giraffe) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Animals Silhouette Solution Tangram Card

Solutions for silhouette outlines of animals (bear, kangaroo, camel, giraffe) made from tangram pieces.…

Solutions for silhouette outlines of animals (fox, rabbit, cat) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Animals Silhouette Solution Tangram Card #2

Solutions for silhouette outlines of animals (fox, rabbit, cat) made from tangram pieces. Tangrams,…

Solutions for silhouette outlines of birds (duck, swan, turkey vulture, cormorant, and egret) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Birds Silhouette Solution Tangram Card

Solutions for silhouette outlines of birds (duck, swan, turkey vulture, cormorant, and egret) made from…

Solutions for silhouette outlines of boats (sailboat, ship, yacht) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Boats Silhouette Solution Tangram Card

Solutions for silhouette outlines of boats (sailboat, ship, yacht) made from tangram pieces. Tangrams,…

Solutions for silhouette outlines of buildings (house, lighthouse) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Buildings Silhouette Solution Tangram Card

Solutions for silhouette outlines of buildings (house, lighthouse) made from tangram pieces. Tangrams,…

Solutions for silhouette outlines of fish (shark, parrot fish, angelfish, flounder, hogfish, beta) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Fish Silhouette Solution Tangram Card

Solutions for silhouette outlines of fish (shark, parrot fish, angelfish, flounder, hogfish, beta) made…

Solutions for silhouette outlines of Florida birds (seagull, mocking bird, egret, flamingo, turkey vulture, cormorant) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Florida Birds Silhouette Solution Tangram Card

Solutions for silhouette outlines of Florida birds (seagull, mocking bird, egret, flamingo, turkey vulture,…

Solutions for silhouette outlines of household items (iron, pipe, kettle, cup, candle, hammer) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Household Items Silhouette Solution Tangram Card

Solutions for silhouette outlines of household items (iron, pipe, kettle, cup, candle, hammer) made…

Solutions for silhouette outlines of people (runners) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

People Silhouette Solution Tangram Card

Solutions for silhouette outlines of people (runners) made from tangram pieces. Tangrams, invented by…

Solutions for silhouette outlines of people (runners, dancers) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

People Silhouette Solution Tangram Card #2

Solutions for silhouette outlines of people (runners, dancers) made from tangram pieces. Tangrams, invented…

Solutions for silhouette outlines of people (woman standing, water skier, woman kneeling, man walking) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

People Silhouette Solution Tangram Card #3

Solutions for silhouette outlines of people (woman standing, water skier, woman kneeling, man walking)…

Solutions for silhouette outlines of people (swimmer, reclining man, butler, diver) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

People Silhouette Solution Tangram Card #4

Solutions for silhouette outlines of people (swimmer, reclining man, butler, diver) made from tangram…

Solutions for silhouette outlines of people (ice skater, saint statue, walking man, reclining man) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

People Silhouette Solution Tangram Card #5

Solutions for silhouette outlines of people (ice skater, saint statue, walking man, reclining man) made…

Solutions for silhouette outlines of people (silhouettes of men) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

People Silhouette Solution Tangram Card #6

Solutions for silhouette outlines of people (silhouettes of men) made from tangram pieces. Tangrams,…

Solutions for silhouette outlines of sea animals (sea monster, dolphin, sea turtle, stingray) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Sea Animals Silhouette Solution Tangram Card

Solutions for silhouette outlines of sea animals (sea monster, dolphin, sea turtle, stingray) made from…

Solutions for silhouette outlines of shapes (rectangle, parallelogram, isosceles triangle, double arrow, rhombus, hexagon) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Shapes Silhouette Solution Tangram Card

Solutions for silhouette outlines of shapes (rectangle, parallelogram, isosceles triangle, double arrow,…

Solutions for silhouette outlines of shapes (quadrilateral, hexagon, arrow, pentagon) made from tangram pieces. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Shapes Silhouette Solution Tangram Card #2

Solutions for silhouette outlines of shapes (quadrilateral, hexagon, arrow, pentagon) made from tangram…

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Tangram Pattern, Black with Small Dotted Lines

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are…

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Tangram Pattern, Black with Solid Lines

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are…

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Tangram Pattern, White with Small Dotted Lines

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are…

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Tangram Pattern, White with Solid Lines

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are…

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Tangram Pattern, Black with Small Dotted Lines and Instructions

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are…

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Tangram Pattern, Black with Solid Lines and Instructions

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are…

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Tangram Pattern, Black with Small Dotted Lines and Instructions

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are…

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are used to develop geometric thinking and spatial sense. 7 figures consisting of triangles, squares, and parallelograms are used to construct the given shapes.

Tangram Pattern, Black with Solid Lines and Instructions

Tangram pattern with black pieces and thin white-dotted lines. Tangrams, invented by the Chinese, are…

A ray has one endpoint and extends indefinitely in the other direction. The ray in this illustration points up and toward the right.

Ray Pointing Up and Right

A ray has one endpoint and extends indefinitely in the other direction. The ray in this illustration…

A ray has one endpoint and extends indefinitely in the other direction. The ray in this illustration points down and toward the left.

Ray Pointing Down and Left

A ray has one endpoint and extends indefinitely in the other direction. The ray in this illustration…