Geometry: Theorems And Constructions

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Geometric construction is part of pure geometry (also known as synthetic geometry or axiomatic geometry). This is the geometry that does not rely on equations and coordinate systems. Instead, it relies on constructions and proofs based on predetermined axioms.

A straightedge is any physical object with a solid, (you guessed it) straight edge that can be traced with a pencil. Note that while many people use a ruler as a straightedge in geometric constructions, technically, a straightedge should not include numbers. Using a ruler is okay as long as you ignore the temptation to compare lines using its measurements.

Geometric constructions (and their accompanying proofs) rely on a certain set of agreed-upon rules called axioms. These essentially give us the tools we need to make a proof and ensure that all readers are working with the same definitions.

If you are given a line, angle, circle, triangle, etc., you can make a copy of it using your straightedge and compass in another place. These constructions will often ask you to put the copy in a specific place, such as a given point or on a given line.

Remember that there are no specific measurements in constructions. That being said, you can copy an angle regardless of the measurement without a protractor by just using a straightedge and compass. You can also create angles of many different measures (for example, 60 degrees, 30 degrees, 75 degrees, etc.) using construction methods.

With a hyperbolic reputation like that, it's little wonder that Pythagoras was credited with devising one of the most famous theorems of all time, even though he wasn't actually the first to come up with the concept. Chinese and Babylonian mathematicians beat him to it by a millennium.

In this lesson, the investigated theorems about triangles have been proven using a variety of methods. Furthermore, with the help of these theorems, the challenge provided at the beginning of the lesson can be solved. Recall the diagram.

This assignment is designed to be completed by 10th grade geometry students. Students will demonstrate their knowledge of geometric constructions by completing practice problems and using varied methods of presenting their work. Choices for presentations include a storyboard, an animation, or a flipbook.

Did you know you can do math without numbers Math is everywhere and with carefully calculated drawing skills, you can create geometric shapes without any numbers at all! These shapes are called geometric constructions and are made from simply a compass and a straight edge.

In Unit 1, Constructions, Proof and Rigid Motion, students are introduced to the concept that figures can be created by just using a compass and straightedge using the properties of circles, and by doing so, properties of these figures are revealed. Transformations that preserve angle measure and distance are verified through constructions and practiced on and off the coordinate plane. These rigid motion transformations are introduced through points and line segments in this unit, and provide the foundation for rigid motion and congruence of two-dimensional figures in Unit 2. This unit lays the groundwork for constructing mathematical arguments through proof and use of precise mathematical vocabulary to express relationships.

Unit 1 begins with students identifying important components to define- emphasizing precision of language and notation as well as appropriate use of tools to represent geometric figures. Students are introduced to the concept of a construction, and use the properties of circles to construct basic geometric figures. In Topic B, students formalize understanding developed in middle school geometry of angles around a point, vertical angles, complementary angles, and supplementary angles through organizing statements and reasons for why relationships to construct a viable argument. Topic C merges the concepts of specificity of definitions, constructions, and proof to formalize rigid motions studied in 8th grade. Students learn that rigid motions can be used as a tool to show congruence. Students focus on rigid motions with points, line segments and angles in this unit through transformation both on and off the coordinate plane.

In the next unit, students use the concepts of constructions, proof, and rigid motions to establish congruence with two dimensional figures. Through the establishment of a solid foundation of precise vocabulary and developing arguments in Unit 1, students are able to use these strategies and theorems to identify and describe geometric relationships throughout the rest of the year.

Later in his life Johnson became interested in mathematics. He was particularly interested in geometry, and most specifically in the problems of antiquity (squaring the circle, trisecting the angle, doubling the cube, and constructing regular polygons). He turned many geometric theorems into works of art. Eighty of his paintings are now at the Smithsonian.

There are geometries in which the ruler is never used to start with. E. g., in finite geometries that only contain a finite number of points and lines, a line is just a (finite) collection of points. On the sphere, the role of straight lines is played by the great circles. The question of geometric construction with the compass alone is not concerned with such kinds of geometries. Geometry of Compass only deals with constructions in the Euclidean plane, and its basic question could be formulated as, What ruler-and-compass constructions could be accomplished with the compass alone

Interestingly, in 1928 the Danish mathematician Hjelmslev discovered in a bookshop in Copenhagen a bookby G. Mohr titled Euclides Danicus (The Danish Euclid) and published in 1672 in Amsterdam. To his great surprise Hjelmslev found a complete treatment of the Mascheroni result in the first part of the book. For this reason, constructions with compass only are commonly referred to as the Mohr-Mascheroni constructions.

Inspired by Mascheroni's result, Jacob Steiner (1796-1863) tried to prove a similar result for astraightedge instead of a compass. In his book Geometrical Constructions Using a Straight Lineand a Fixed Circle published in 1833, Steiner was able to prove that given a fixed circle and its center, all the constructions in the plane can be carried out by the straightedge alone. Using only elementary Projective Geometry itcan be shown that the center of the circle is indispensable.

The difficulty obviously lies with the last two problems. In the Geometry of Compassconstructions may be awfully obscure even for simple problems. To avoid complicating the matters it'salways useful to split a problem into a number of simpler steps. A proof to the Mascheroni result willemerge as a combination of the problems below. (However, not all of the problems are related to theproof.)

We will make basic hyperbolic tools as GeoGebra constructions and then use them to make basic hyperbolic ruler and compass constructions. To make it easy, the unit circle centered at the origin will be used as \\(C_\\infty \\) in all constructions.

In the static case, where the points do not move, it is fairly easy to make hyperbolic constructions. In the dynamic case however, the constructions should pass the dragging test. For that reason it is often necessary to use the GeoGebra If-command and the GeoGebra IsDefined-command.

If a unit circle is created and then used when constructing the geodesic, it is not possible to make a GeoGebra tool without using the unit circle as input object. For that reason all constructions using the unit circle should be entered in the input bar.

Since the proofs of the congruent triangle criterions and the isosceles triangle theorem do not use the parallel postulate, the two constructions must also hold in non-Euclidean geometry, as long as the non-Euclidean ruler and compass have the same functionality as Euclidean ruler and compass.

When making the hyperbolic constructions you will need hyperbolic tools for making a: line, segment, ray, distance, and circle. You can either make them yourself by following the instructions in the text or copy the activity Basic hyperbolic tools. You must also use the regular GeoGebra tools: Point, Intersect, and Reflect about Circle.

N2 - The algebra of densities $\\Den(M)$ is a commutative algebra canonically associated with a given manifold or supermanifold $M$. We introduced this algebra earlier in connection with our studies of Batalin--Vilkovisky geometry. The algebra $\\Den(M)$ is graded by real numbers and possesses a natural invariant scalar product. This leads to important geometric consequences and applications to geometric constructions on the original manifold. In particular, there is a classification theorem for derivations of the algebra $\\Den(M)$.It allows {a natural definition of}bracket operations on vector densities of various weights on a (super)manifold $M$,similar to how the classical Fr\\\"{o}licher--Nijenhuis theorem on derivations of the algebra of differential forms leads to the Nijenhuis bracket. It is possible to extend this classification from ``vector fields'' (derivations) on $\\Den(M)$ to ``multivector fields''. This leads to the striking result that an arbitrary even Poisson structure on $M$ possesses a canonical lifting to the algebra of densities. (The latter two statements were obtained by our student A.Biggs.) This is in sharp contrast with the previously studied case of an odd Poisson structure, where extra data are required for such a lifting.

AB - The algebra of densities $\\Den(M)$ is a commutative algebra canonically associated with a given manifold or supermanifold $M$. We introduced this algebra earlier in connection with our studies of Batalin--Vilkovisky geometry. The algebra $\\Den(M)$ is graded by real numbers and possesses a natural invariant scalar product. This leads to important geometric consequences and applications to geometric constructions on the original manifold. In particular, there is a classification theorem for derivations of the algebra $\\Den(M)$.It allows {a natural definition of}bracket operations on vector densities of various weights on a (super)manifold $M$,similar to how the classical Fr\\\"{o}licher--Nijenhuis theorem on derivations of the algebra of differential forms leads to the Nijenhuis bracket. It is possible to extend this classification from ``vector fields'' (derivations) on $\\Den(M)$ to ``multivector fields''. This leads to the striking result that an arbitrary even Poisson structure on $M$ possesses a canonical lifting to the algebra of densities. (The latter two statements were obtained by our student A.Biggs.) This is in sharp contrast with the previously studied case of an odd Poisson structure, where extra data are required for such a lifting. 59ce067264