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This figure represents an apparatus frequently employed for illustrating some of the consequences of Torricelli's Theorem.

Apparatus for verifying Torricelli's Theorem

This figure represents an apparatus frequently employed for illustrating some of the consequences of…

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 "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 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…

An illustration used to show how the area of a circle is calculated. Area is equal to the product of pi and the radius squared.

Area Of A Circle

An illustration used to show how the area of a circle is calculated. Area is equal to the product of…

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 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 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 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 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 prove the corollary that "Two lines perpendicular respectively to two intersecting lines also intersect."

Intersecting Lines Corollary

Illustration used to prove the corollary that "Two lines perpendicular respectively to two intersecting…

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

Perpendicular to Line Corollary

Illustration used to prove the corollary that "From a point outside a line there exists only one perpendicular…

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.

Model Of Geometric Proportions In A Circle

An illustration showing a model of a circle with intersecting chords that illustrates the following…

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.

Model Of Geometric Proportions In A Triangle

An illustration showing a model of a triangle that illustrates the following relationship: a:c = d:(b…

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.

Model Of Geometric Proportions

An illustration showing a quadrilateral model that illustrates the following relationships: a:b = c:d,…

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.

Model Of Geometric Proportions

An illustration showing a model that illustrates the following relationships: a:b = c:d, ad = bc. Product…

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

Model Of Geometric Proportions

An illustration showing a model that illustrates the following relationships: a:b = c:b, ab = c²,…

An illustration showing a model that illustrates the following relationship: A:B = a:b.

Model Of Geometric Proportions

An illustration showing a model that illustrates the following relationship: A:B = a:b.

An illustration showing a model that illustrates the following relationships: a:x = x:a - x, x = √(a&sup2 + (a/2)&sup2 - a/2).

Model Of Geometric Proportions

An illustration showing a model that illustrates the following relationships: a:x = x:a - x, x = √(a²…

An illustration showing a model of 2 circles with tangent lines, diameters, and radii that illustrates the following geometric relationship: "x = aR/(R - r), a = √(t&sup2 + (R - r)&sup2), t = √(a&sup2 - (R - r)&sup2, sin.v = t/a."

Model Of Geometric Relationships In 2 Circles

An illustration showing a model of 2 circles with tangent lines, diameters, and radii that illustrates…

An illustration showing a model of 2 circles with tangent lines, diameters, and radii that illustrates the following geometric relationship: " t = √(a&sup2 - (R + r)&sup2, a = √(t&sup2 - (R + r)&sup2 "

Model Of Geometric Relationships In 2 Circles

An illustration showing a model of 2 circles with tangent lines, diameters, and radii that illustrates…

An illustration showing a model of a circle with an exterior angle formed between a tangent and a secant that illustrates the following geometric relationship: a:t = t:b, t&sup2 = ab

Model Of Geometric Relationships In A Circle

An illustration showing a model of a circle with an exterior angle formed between a tangent and a secant…

An illustration showing a model of a circle with angles formed between tangents and secants that illustrates the following geometric relationship: t&sup2 = (a + b)(a - b).

Model Of Geometric Relationships In A Circle

An illustration showing a model of a circle with angles formed between tangents and secants that illustrates…

"If from the foot of a perpendicular to a plane a straight line is drawn at right angles to any line in the lane, and its intersection with that line is joined to any point of the perpendicular, this last line will be perpendicular to the line in the plane."

Perpendicular To Line In Plane

"If from the foot of a perpendicular to a plane a straight line is drawn at right angles to any line…

"All the perpendiculars to a straight line at the same point lie in a plane perpendicular to the line."

Plane Perpendicular To Line

"All the perpendiculars to a straight line at the same point lie in a plane perpendicular to the line."

"Two planes perpendicular to the same straight line are parallel."

Planes Perpendicular To A Line

"Two planes perpendicular to the same straight line are parallel."

"Two straight lines perpendicular to the same plane are parallel."

Parallel Lines

"Two straight lines perpendicular to the same plane are parallel."

"Two straight lines that are parallel to a third are parallel to each other."

Parallel Lines

"Two straight lines that are parallel to a third are parallel to each other."

"If a straight line is perpendicular to each of two straight lines at their point of intersection it is perpendicular to the plane of those lines."

Line Perpendicular To Plane

"If a straight line is perpendicular to each of two straight lines at their point of intersection it…

"If one of two parallels is perpendicular to a plane, the other is also."

Lines Perpendicular To A Plane

"If one of two parallels is perpendicular to a plane, the other is also."

Illustration used to prove "Two angles whose sides are perpendicular each to each are either equal or supplementary."

Angles With Perpendicular Sides Are Equal or Supplementary Proof

Illustration used to prove "Two angles whose sides are perpendicular each to each are either equal or…

Illustration used to prove the Pythagorean Theorem, according to Euclid. A perpendicular is drawn from the top vertex of the right triangle extended through the bottom square, forming 2 rectangles. Each rectangle has the same area as one of the two legs. This proves that the sum of the squares of the legs is equal to the square of the hypotenuse (Pythagorean Theorem).

Euclid's Pythagorean Theorem Proof

Illustration used to prove the Pythagorean Theorem, according to Euclid. A perpendicular is drawn from…

Illustration used to prove "An exterior angle of a triangle is equal to the sum of the two remote interior angles."

Exterior Angle Proof

Illustration used to prove "An exterior angle of a triangle is equal to the sum of the two remote interior…

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is equal to the square of the hypotenuse.

Geometric Pythagorean Theorem Proof

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is…

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is equal to the square of the hypotenuse. The geometrical illustration depicts a 3,4,5 right triangle with the square units drawn to prove that the sum of the squares of the legs (9 + 16) equals the square of the hypotenuse.

Geometric Pythagorean Theorem Proof

Illustration that can be used to prove the Pythagorean Theorem, the sum of the squares of the legs is…

A visual illustration used to prove the Pythagorean Theorem by rearrangement. When the 4 identical triangles are removed, the areas are equal. Thus, proving the sum of the squares of the legs is equal to the square of the hypotenuse.

Pythagorean Theorem Proof by Rearrangement

A visual illustration used to prove the Pythagorean Theorem by rearrangement. When the 4 identical triangles…

Illustration used to prove "If the sides of any polygon are prolonged in succession one way, no two adjacent sides being prolonged through the same vertex, the sum of the exterior angles thus formed is four right angles."

Sum of Exterior Angles of a Polygon Proof

Illustration used to prove "If the sides of any polygon are prolonged in succession one way, no two…

Illustration used to prove "The sum of all the angles of any polygon is twice as many right angles as the polygon has sides, less four right angles."

Sum of Interior Angles of a Polygon Proof

Illustration used to prove "The sum of all the angles of any polygon is twice as many right angles as…

"A proof-plane may be made by cementing a bronze cent or a disk of gilt paper to a thin insulating handle, as a glass tube or a vulcanite rod. Slide the disk of the proof-plane along the surface of the electrified body to be tested, and quickly bring it into contact with the knob of the gold-leaf electroscope, the leaves of which will diverge." — Avery, 1895

Hand with Proof-plane

"A proof-plane may be made by cementing a bronze cent or a disk of gilt paper to a thin insulating handle,…

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides of the other, but the included angle of the first greater than the included angle of the second, then the third side of the first is greater than the third side of the second."

2 Triangles Theorem

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides…

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides of the other, but the third side of the first greater than the third side of the second, then the angle opposite the third side of the first is greater than the angle opposite the third side of the second."

2 Triangles Theorem

Illustration used to prove that "If two triangles have two sides of one equal respectively to two sides…

Illustration used to prove that "If one side of a triangle is prolonged, the exterior angle formed is greater than either of the remote interior angles."

Exterior Angle of Triangle Theorem

Illustration used to prove that "If one side of a triangle is prolonged, the exterior angle formed is…

Illustration used to prove that "If two straight lines are parallel to a third straight line, they are parallel to each other."

Parallel Lines Theorem

Illustration used to prove that "If two straight lines are parallel to a third straight line, they are…

Illustration used to prove that "If two sides of a triangle are unequal, the angle opposite the greater side is greater than the angle opposite the less side."

Sides of Triangle Theorem

Illustration used to prove that "If two sides of a triangle are unequal, the angle opposite the greater…

Illustration used to prove that "The sum of any two sides of a triangle is greater than the third side."

Sides of Triangle Theorem

Illustration used to prove that "The sum of any two sides of a triangle is greater than the third side."

Illustration of triangle ABC with BE extended through the triangle at point D. Segment AB is equal to segment BD.

Segments Labeled In A Triangle

Illustration of triangle ABC with BE extended through the triangle at point D. Segment AB is equal to…