Theorems and Proofs - Clipart ETC

Theorems and Proofs

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Clipart images that can be used to demonstrate various geometric theorems and proofs.

Angle, Face Angles of Convex Polyhedral
Diagram used to prove the theorem: "The sum of the face angles of any convex polyhedral angle is less than four right angles."...

Angles, Equal Right
Illustration used to prove that all right angles are equal....

Angles, Symmetrical or Equal Trihedral
Diagram used to prove the theorem: "Two trihedral angles, which have three face angles of the one equal respectively to three face angles of the other , are either equal or symmetrical."...

Binomial (a + b), Squaring
An illustration showing how to square binomial (a + b). (a + b)² = a² + 2ab + b²....

Binomial (a - b), Squaring
An illustration showing how to square binomial (a - b). (a - b)² = a² - 2ab + b²....

Circle Diameters
If every diameter is perpendicular to its conjugate, the conic is a circle....

Circular Involution
Every elliptical involution has the property that there are two definite points in the plane from which any two conjugate points are seen under a right angle....

Corollary, Intersecting Lines
Illustration used to prove the corollary that "Two lines perpendicular respectively to two intersecting lines also intersect."...

Corollary, Perpendicular to Line
Illustration used to prove the corollary that "From a point outside a line there exists only one perpendicular to the line."...

Cross-Ratio
The cross-ratio of four points in a line is equal to the cross-ratio of their projections on any other line which lies in the same plane with it....

Curve Tangents
Any four-point on a curve of the second order and the four-side formed by the tangents at these points stand in this relation that the diagonal points of the four-point lie in the diagonals of the fou...

Dedekind Property 1
Illustration 1 of the Dedekind axiom....

Dedekind Property 2
Illustration 2 of the Dedekind axiom....

Dedekind Property 3
Illustration 3 of the Dedekind axiom....

Dedekind Property 4
Illustration 4 of the Dedekind axiom....

Dihedral Angle, Plane Bisecting
Diagram used to prove the theorem: "Every point in a plane which bisects a dihedral angle is equidistant from the faces of the angle"...

Equilateral Triangle, Equilateral Triangle Inscribed In
Illustration used to prove that triangle EFD is equilateral given that triangle ABC is equilateral and AE=BF=CD....

Frustum of a Pyramid, Volume of
Diagram used to prove the theorem: "The volume of the frustum of a pyramid (cone) is equal to the sum of three pyramids (cones) whose common altitude is the altitude of the frustum and whose bases are...

Geometric Proportions In A Circle, Model Of
An illustration showing a model of a circle with intersecting chords that illustrates the following relationship: a:c = b:d, ad = bc. Product of the means equals the product of the extremes....

Geometric Proportions In A Triangle, Model Of
An illustration showing a model of a triangle that illustrates the following relationship: a:c = d:(b - d), d = (a × b) ÷ (c + a), v = v. ...

Geometric Proportions, Model Of
An illustration showing a quadrilateral model that illustrates the following relationships: a:b = c:d, ad = bc, A = B. Product of the means equals the product of the extremes....

Geometric Proportions, Model Of
An illustration showing a model that illustrates the following relationships: a:b = c:d, ad = bc. Product of the means equals the product of the extremes....

Geometric Proportions, Model Of
An illustration showing a model that illustrates the following relationships: a:b = c:b, ab = c², c = √ab....

Geometric Proportions, Model Of
An illustration showing a model that illustrates the following relationship: A:B = a:b....

Geometric Proportions, Model Of
An illustration showing a model that illustrates the following relationships: a:x = x:a - x, x = √(a² + (a/2)² - a/2)....

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Clipart ETC is a part of the Educational Technology Clearinghouse and is funded by various grants. Produced by the Florida Center for Instructional Technology, College of Education, University of South Florida. Email the project manager.

	Angle, Face Angles of Convex Polyhedral Diagram used to prove the theorem: "The sum of the face angles of any convex polyhedral angle is less than four right angles."...
	Angles, Equal Right Illustration used to prove that all right angles are equal....
	Angles, Symmetrical or Equal Trihedral Diagram used to prove the theorem: "Two trihedral angles, which have three face angles of the one equal respectively to three face angles of the other , are either equal or symmetrical."...
	Binomial (a + b), Squaring An illustration showing how to square binomial (a + b). (a + b)² = a² + 2ab + b²....
	Binomial (a - b), Squaring An illustration showing how to square binomial (a - b). (a - b)² = a² - 2ab + b²....
	Circle Diameters If every diameter is perpendicular to its conjugate, the conic is a circle....
	Circular Involution Every elliptical involution has the property that there are two definite points in the plane from which any two conjugate points are seen under a right angle....
	Corollary, Intersecting Lines Illustration used to prove the corollary that "Two lines perpendicular respectively to two intersecting lines also intersect."...
	Corollary, Perpendicular to Line Illustration used to prove the corollary that "From a point outside a line there exists only one perpendicular to the line."...
	Cross-Ratio The cross-ratio of four points in a line is equal to the cross-ratio of their projections on any other line which lies in the same plane with it....
	Curve Tangents Any four-point on a curve of the second order and the four-side formed by the tangents at these points stand in this relation that the diagonal points of the four-point lie in the diagonals of the fou...
	Dedekind Property 1 Illustration 1 of the Dedekind axiom....
	Dedekind Property 2 Illustration 2 of the Dedekind axiom....
	Dedekind Property 3 Illustration 3 of the Dedekind axiom....
	Dedekind Property 4 Illustration 4 of the Dedekind axiom....
	Dihedral Angle, Plane Bisecting Diagram used to prove the theorem: "Every point in a plane which bisects a dihedral angle is equidistant from the faces of the angle"...
	Equilateral Triangle, Equilateral Triangle Inscribed In Illustration used to prove that triangle EFD is equilateral given that triangle ABC is equilateral and AE=BF=CD....
	Frustum of a Pyramid, Volume of Diagram used to prove the theorem: "The volume of the frustum of a pyramid (cone) is equal to the sum of three pyramids (cones) whose common altitude is the altitude of the frustum and whose bases are...
	Geometric Proportions In A Circle, Model Of An illustration showing a model of a circle with intersecting chords that illustrates the following relationship: a:c = b:d, ad = bc. Product of the means equals the product of the extremes....
	Geometric Proportions In A Triangle, Model Of An illustration showing a model of a triangle that illustrates the following relationship: a:c = d:(b - d), d = (a × b) ÷ (c + a), v = v. ...
	Geometric Proportions, Model Of An illustration showing a quadrilateral model that illustrates the following relationships: a:b = c:d, ad = bc, A = B. Product of the means equals the product of the extremes....
	Geometric Proportions, Model Of An illustration showing a model that illustrates the following relationships: a:b = c:d, ad = bc. Product of the means equals the product of the extremes....
	Geometric Proportions, Model Of An illustration showing a model that illustrates the following relationships: a:b = c:b, ab = c², c = √ab....
	Geometric Proportions, Model Of An illustration showing a model that illustrates the following relationship: A:B = a:b....
	Geometric Proportions, Model Of An illustration showing a model that illustrates the following relationships: a:x = x:a - x, x = √(a² + (a/2)² - a/2)....