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Illustration showing that can be used to prove that the base angles of an isosceles triangle are equal.

Base Angles In An Isosceles Triangle

Illustration showing that can be used to prove that the base angles of an isosceles triangle are equal.

Illustration showing that if equal segments measured from the end of the base are laid off on the base of an isosceles triangle, the lines joining the vertex of the triangle to the ends of the segments will be equal.

Equal Segments In An Isosceles Triangle

Illustration showing that if equal segments measured from the end of the base are laid off on the base…

Illustration showing that if equal segments measured from the end of the base prolonged are laid off on the base of an isosceles triangle, the lines joining the vertex of the triangle to the ends of the segments will be equal.

Equal Segments In An Isosceles Triangle

Illustration showing that if equal segments measured from the end of the base prolonged are laid off…

Illustration used to prove that triangle EFD is equilateral given that triangle ABC is equilateral and AE=BF=CD.

Equilateral Triangle Inscribed In An Equilateral Triangle

Illustration used to prove that triangle EFD is equilateral given that triangle ABC is equilateral and…

Illustration that uses triangles to measure the distance across a lake.

Using Triangles To Measure A Lake

Illustration that uses triangles to measure the distance across a lake.

Illustration that uses triangles to measure the distance across a river.

Using Triangles To Measure A River

Illustration that uses triangles to measure the distance across a river.

Illustration that uses triangles to measure the distance across a river.

Using Triangles To Measure A River

Illustration that uses triangles to measure the distance across a river.

Illustration that uses triangles to measure the distance across a river.

Using Triangles To Measure A River

Illustration that uses triangles to measure the distance across a river.

Illustration that uses triangles to measure the distance across a lake.

Using Triangles To Measure A Lake

Illustration that uses triangles to measure the distance across a lake.

Illustration used to show that two triangles are equal if the three sides of one are equal respectively to the three sides of the other.

Equal Triangles

Illustration used to show that two triangles are equal if the three sides of one are equal respectively…

Illustration used to show how to construct an equilateral triangle, with a given line as a side.

Construction Of Equilateral Triangle

Illustration used to show how to construct an equilateral triangle, with a given line as a side.

Illustration showing the three angle bisectors in a triangle.

Angle Bisectors In A Triangle

Illustration showing the three angle bisectors in a triangle.

Illustration showing a perpendicular bisector of a triangle extended outside of the triangle.

Triangle With Perpendicular Bisector

Illustration showing a perpendicular bisector of a triangle extended outside of the triangle.

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 of a triangle with interior segments and angles labeled.

Segments and Angles in a Triangle

Illustration of a triangle with interior segments and angles labeled.

Illustration of a triangle with interior segments and angles labeled.

Segments and Angles in a Triangle

Illustration of a triangle with interior segments and angles labeled.

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 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 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…

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 of the construction used to create an isosceles triangle, given the bases and the sum of the altitude and a side.

Construction Of An Isosceles Triangle

Illustration of the construction used to create an isosceles triangle, given the bases and the sum of…

Illustration used to prove the theorem, "The sum of the angles of any triangle is two right angles."

Sum Of Angles In Triangle Theorem

Illustration used to prove the theorem, "The sum of the angles of any triangle is two right angles."

"A circle may be considered as made up of triangles whose bases form the circumference, and whose altitude is the radius (1/2 diameter) of the circle." This is clearly shown by the cut at the left.

Circle Made Up Of Triangles

"A circle may be considered as made up of triangles whose bases form the circumference, and whose altitude…

A triangular stem.

Triangular Stem

A triangular stem.

A triangular leaf.

Triangular Leaf

A triangular leaf.

A trowel-shaped leaf.

Trowel-Shaped Leaf

A trowel-shaped leaf.

"In the triangle above, the line AB is its altitude. Since we know how to find the area of one triangle, we can find the areas of as many triangles as we have made from our circle. Therefore, to find the area of a circle: Find the area of one of the triangles and multiply by the number of triangles." -Foster, 1921

Area of Circle with Triangles

"In the triangle above, the line AB is its altitude. Since we know how to find the area of one triangle,…

An illustration showing an isosceles triangle in an equilateral triangle.

Equilateral And Isosceles Triangles

An illustration showing an isosceles triangle in an equilateral triangle.

An illustration showing how to construct a center and radius of a circle that will tangent the three sides of a triangle. "Bisect two of the angles in the triangle, and the crossing C is the center of the required circle."

Construction Of The Center And Radius Of A Circle Tangent To Triangle Sides

An illustration showing how to construct a center and radius of a circle that will tangent the three…

An illustration showing how to construct an equilateral triangle inscribed in a circle. "With the radius of the circle and center C draw the arc DFE; with the same radius, and D and E as centers, set off the points A and B. Join A and B, B and C, C and A, which will be the required triangle."

Construction Of An Equilateral Triangle Inscribed In A Circle

An illustration showing how to construct an equilateral triangle inscribed in a circle. "With the radius…

An illustration showing that the area of a regular polygon is equal to the area of a triangle whose base is equal to the sum of all the sides, and the height a equal to the appotem of the polygon. "The reason of this is that the area of two or more triangles ABC and ADC having a common or equal base b and equal height h are alike."

Area Of Regular Polygon Proof

An illustration showing that the area of a regular polygon is equal to the area of a triangle whose…

An illustration showing how to construct an ellipse using a string. "Having given the two axes, set off from c half the great axis at a and b, which are the two focuses of the ellipse. Take an endless string as long as the three sides in the triangle abc, fix two pins or nails in the focuses, one in a and one in b, lay the string around a and b, stretch it with a pencil d, which then will describe the desired ellipse."

Construction Of An Ellipse

An illustration showing how to construct an ellipse using a string. "Having given the two axes, set…

An illustration showing a triangle with sides b, c, and d inscribed in a circle with radius r.

Triangle Inscribed In A Circle

An illustration showing a triangle with sides b, c, and d inscribed in a circle with radius r.

An illustration showing a circle with radius r inscribed in a triangle with sides a, b, and c.

Circle Inscribed In A Triangle

An illustration showing a circle with radius r inscribed in a triangle with sides a, b, and c.

An illustration showing a triangle with interior angles A, B, C, and exterior angles D, and A' + B'.

Exterior And Interior Angles Of A Triangle

An illustration showing a triangle with interior angles A, B, C, and exterior angles D, and A' + B'.

An illustration showing a triangle with angles A, C, and D inscribed in a circle which is tangent to a line.

Triangle Inscribed In A Circle

An illustration showing a triangle with angles A, C, and D inscribed in a circle which is tangent to…

An illustration showing a circle with sectors D and E inscribed in a triangle with angles, A, B, and C.

Circle Inscribed In A Triangle

An illustration showing a circle with sectors D and E inscribed in a triangle with angles, A, B, and…

An illustration showing a model that illustrates the Pythagorean Theorem: a&sup2 + b&sup2 = c&sup2.

Model Of Pythagorean Theorem

An illustration showing a model that illustrates the Pythagorean Theorem: a² + b² = c².

An illustration of an acute triangle with the height/altitude labeled h.

Acute Triangle

An illustration of an acute triangle with the height/altitude labeled h.

An illustration of an obtuse triangle with the height/altitude labeled h.

Obtuse Triangle

An illustration of an obtuse triangle with the height/altitude labeled h.

An illustration of a right triangle inscribed in a semicircle.

Right Triangle Inscribed In A Semicircle

An illustration of a right triangle inscribed in a semicircle.

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 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…

"The triangle ABC is divided into 2 right triangles I and II. ABC is seen to be equal to 1/2 of the rectangle ABNO." -Foster, 1921

Area of Triangle

"The triangle ABC is divided into 2 right triangles I and II. ABC is seen to be equal to 1/2 of the…

"The child sees triangle ACB = triangle ADB, and that I + II = CA - DB; and so he sees that the area of triangle=1/2 area of rectangle whose base and altitude are the same as those of the triangles." -Foster, 1921

Area of Triangle

"The child sees triangle ACB = triangle ADB, and that I + II = CA - DB; and so he sees that the area…

"Fig. 3 shows triangle I = triangle II, III = IV, and so triangle ABC = 1/2 of rectangle ABDE. The fact is realized that the area of a triangle equals 1/2 the product of the base and altitude." -Foster, 1921

Area of Triangle

"Fig. 3 shows triangle I = triangle II, III = IV, and so triangle ABC = 1/2 of rectangle ABDE. The fact…

Illustration showing how to find the area of a hexagon using the triangles that make it up.

Area of Hexagon

Illustration showing how to find the area of a hexagon using the triangles that make it up.

"Diagram to illustrate the inferred structure in the vicinity of the "Triangle." Arkose conglomerate is represented by lines and ellipses, anterior basalt by the black areas, anterior shale by parallel lines, and main basalt by cross-hachure areas. The finer hachuring (and hence the deeper shade) corresponds to the higher blocks of the basalt." -Walcott, 1901

Basalt Diagram

"Diagram to illustrate the inferred structure in the vicinity of the "Triangle." Arkose conglomerate…

The region of the neck, from the side.

Side View of Neck

The region of the neck, from the side.

Section of the aorta, to show the action of the semilunar valve. A is intended to show the valves, represented by the dotted lines, lying near the arterial walls, represented by the continuous outer line. B (after Hunter) shows the arterial wall distended into three pouched (a), and drawn away from the valves, which are straightened into the form of an equilateral triangle, as represented by the dotted line.

Action of Semilunar Valve

Section of the aorta, to show the action of the semilunar valve. A is intended to show the valves, represented…

This modern Tablet Architectural frame was in the style of the Italian Renaissance. It had the general shape of an erect triangle that has a cresting feature, free-ending upwards.

Tablet Frame

This modern Tablet Architectural frame was in the style of the Italian Renaissance. It had the general…

The pulpit Architectural frame was a German frame that was dated between 1595 to 1597. It had the general shape of an erect triangle that has a cresting feature, free-ending upwards.

Pulpit Frame

The pulpit Architectural frame was a German frame that was dated between 1595 to 1597. It had the general…

Labeled shapes showing various measures of form.

Measures of Form

Labeled shapes showing various measures of form.

Showing different types of forms or shapes: rectangle, right triangle, acute triangle, and obtuse triangle.

Forms

Showing different types of forms or shapes: rectangle, right triangle, acute triangle, and obtuse triangle.

Four different types of forms or shapes: right triangle, isosceles triangle, rectangle, and circle.

Shapes

Four different types of forms or shapes: right triangle, isosceles triangle, rectangle, and circle.

Illustration showing that a circle may be considered as made up of triangles whose bases form the circumference.

Triangles Making Up A Circle

Illustration showing that a circle may be considered as made up of triangles whose bases form the circumference.

Illustration of a triangular prism; a prism whose bases are triangles.

Triangular Prism

Illustration of a triangular prism; a prism whose bases are triangles.

A cylindrical helix is a curve generated by a point moving uniformly around a cylinder and uniformly lengthwise of the cylinder at the some time. The hypotenuse of a right triangle will form one turn of a helix if it is wrapped around a cylinder. The base of the triangle is equal to the circumference of the cylinder and the altitude is the pitch of the helix.

Helix

A cylindrical helix is a curve generated by a point moving uniformly around a cylinder and uniformly…

An apparatus for verifying the parallelogram of forces. This method is used in physics for determining the force of vectors.

Gravesande's Apparatus

An apparatus for verifying the parallelogram of forces. This method is used in physics for determining…