What is Plane geometry: Definition and 16 Discussions
Euclidean geometry is a mathematical system attributed to Alexandrian Greek mathematician Euclid, which he described in his textbook on geometry: the Elements. Euclid's method consists in assuming a small set of intuitively appealing axioms, and deducing many other propositions (theorems) from these. Although many of Euclid's results had been stated by earlier mathematicians, Euclid was the first to show how these propositions could fit into a comprehensive deductive and logical system. The Elements begins with plane geometry, still taught in secondary school (high school) as the first axiomatic system and the first examples of mathematical proofs. It goes on to the solid geometry of three dimensions. Much of the Elements states results of what are now called algebra and number theory, explained in geometrical language.For more than two thousand years, the adjective "Euclidean" was unnecessary because no other sort of geometry had been conceived. Euclid's axioms seemed so intuitively obvious (with the possible exception of the parallel postulate) that any theorem proved from them was deemed true in an absolute, often metaphysical, sense. Today, however, many other self-consistent non-Euclidean geometries are known, the first ones having been discovered in the early 19th century. An implication of Albert Einstein's theory of general relativity is that physical space itself is not Euclidean, and Euclidean space is a good approximation for it only over short distances (relative to the strength of the gravitational field).Euclidean geometry is an example of synthetic geometry, in that it proceeds logically from axioms describing basic properties of geometric objects such as points and lines, to propositions about those objects, all without the use of coordinates to specify those objects. This is in contrast to analytic geometry, which uses coordinates to translate geometric propositions into algebraic formulas.
Currently, as far as I know, the two main ways to express any given point on a plane is through either cartesian plane or polar coordinates. Both of which requires an ordered pair of two numbers to express a point. However, I wonder if there exists such a system that could express any given...
Before looking at the proof of basic theorems in Euclidean plane geometry, I feel that I should draw pictures or use other physical objects to have some idea why the theorem must be true. After all, I should not just plainly play the "game of logic". And, it is from such observations in real...
Homework Statement
Describe and sketch the geometric objects represented by the
systems of equations
Homework Equations
x2 + y2 + z2 = 4
x + y + z = 1
The Attempt at a Solution
I can sketch both objects:
1) sphere with center (0,0,0) and radius 2
2) "simple" plane with intersection...
Mod note: Moved from a technical forum section, so missing the homework template.
@fab13 -- please post homework problems in the appropriate section under Homework & Coursework.
I have the following exercise to solve : I have to find all the points on the surface ##x^2+y^2+z^2=36## (so a sphere...
Hello,
I am totally bad at geometry , by geometry I mean plane euclidean geometry with similarities and circles. I sometimes feel totally lost with problems. For example:
The parallel sides of trapezoid ABCD are 3 cm and 9 cm(AB and DC).The non parallel sides are 4 cm and 6 cm(AD and BC).A...
Homework Statement
Problem 99 from "Kiselev's Geometry Book I - Planimetry":
Two isosceles triangles with a common vertex and congruent lateral sides cannot fit one inside the other.
Homework EquationsThe Attempt at a Solution
The statement is obviously true. If we visualize each isosceles...
Homework Statement
Problem 55 from Kiselevś Geometry - Book I. Planimetry: "Prove that each diagonal of a quadrilateral either lies entirely in its interior, or entirely in its exterior. Give an example of a pentagon for which this is false."
Homework EquationsThe Attempt at a Solution
The...
Hello,
After reading both How to Prove It: A Structured Approach - By Daniel J Velleman, and one of the Lost Feynman Lectures on Planetary Orbits, I'm wondering if anyone could suggest to me any good books they've read (or heard about) pertaining to logic (paired with analysis), or plane...
Homework Statement
Find the equation of plane which contains the line l:\left\{\begin{array}{l} x=t+2 \\y=2t-1\\z=3t+3 \end{array}\right., and makes the angle of \frac{2\pi}{3} with the plane \pi:x+3y-z+8=0.
The Attempt at a Solution
My attempt was to find the normal vector of plane which...
Consider a point P inside a triangle ABC. Angle PBC is 10 degrees, angle PCB is 20 degrees, and angle BAC is 100 degrees. Find angle PAC.
Question is that is this problem even solvable? I found it in an Olympiad training book...
Hi everyone,
Can anyone solve the followong by plane Euclidean geometry?
I got it by co-ordinate geometry, but couldn't get it by plane...
>In an acute - angled triangle PQR , angle P=\pi/6 , H is the orthocentre, and M is the midpoint of QR . On the line HM , take a point T such that...
Hey there
I do study a textbook at School called Plane Geometry, it is about 3d dimensional figures, it starts with explaining Planes, points and lines, then It starts with theories, then I have to prove something like a line perpendicular to a plane and things like that. I googled Plane...
This is the last problem on a geometry problem set that I can't seem to finish.
AB and BC are chords in a circle where AB > BC. D is the midpoint of minor arc ADBC. If DE is perpendicular to AB, prove that AE = EB + BC.
I would really appreciate just the proper way to approach this...
PQR is an equilateral triangle, and A is a point on QR such that RA = 2 QA. Prove that PA^2 = 7QA^2
If someone can help, i'd really appreciate it. Thanks