Mathematics Stack Exchange
Mathematics Stack Exchange

Questions tagged [euclidean-geometry]

For questions on geometry assuming Euclid's parallel postulate.

The geometry of Euclid is based on five axioms (Euclid called them postulates). Any geometry based on the first four of these is called an absolute geometry. The fifth one states:

If a line segment intersects two straight lines forming two interior angles on the same side that sum to less than two right angles, then the two lines, if extended indefinitely, meet on that side on which the angles sum to less than two right angles.

It was observed by Proclus that, in the presence of the the other four postulates, Euclid's fifth postulate can be replaced by Playfair's axiom:

Given a line and a point not on it, then one and only one line parallel to the given line can be drawn through the point.

The independence of the parallel postulate and its equivalent formulations from the first four axioms was shown by Beltrami in 1868.

Another alternative definition is that two lines are parallel if every perpendicular extended from one meets the other as a perpendicular.

9328 questions
0
votes
1 answer

isosceles triangles and their perpendiculars proof

Let an isosceles triangle ABC in the Euclidean plane be given, with AB being the base. DRAW the perpendiculars AD and BE from A and B onto BC and AC. Show that
Elizabeth Hill
  • 37
0
votes
1 answer

Why are values squared in distance formulas, such as the Pythagorean Theorem?

Why do you square the values in the Pythagorean Theorem or any distance formula wherein you're trying to find the distance between two points in two-dimensional, Euclidean space? for example, why are we squaring the difference in the two $x,x$…
Jennifer Melfi
  • 1
0
votes
1 answer

Geometric Construction Rhombus

Given two line segments, Construct a rhombus whose diagonals have lengths equal to the lengths of the two given segments. I can get to finding perpendicular bisectors of each line segment, but have no idea how to move one so that it lies over the…
Jacob Rodgers
  • 477
0
votes
0 answers

Euclidean geometry with ruler and compass

I was wondering, is there any book out there that is of the style of Euclid's Elements ? One which you have to use a compass and ruler for certain propositions like building a triangle, etc. Or would it be better to study the elements directly ?…
user108343
0
votes
1 answer

A quadrilateral with one pair of opposite right angles. Is this a rectangle?

I can prove it's not a rectangle by drawing some lines, but is there a name for this kind of figure? Thanks.
user232751
  • 99
0
votes
1 answer

Non-Euclidean geometries

Does non-Euclidean geometry can be always immersed in Euclidean of dimension D+1? This is probably very basic question, but I'm just trying to understand why do you need to consider sometimes very complicated non-euclidean geometries, as for example…
WeSenseASoulInSearchOfAnswers
  • 131
0
votes
1 answer

Reduce distance between two points by %

I have two points, say A = (2, 6) and B = (5, 3). I want to move point B up to 70% closer to point A. I calculate Euclidean distance between two points - it is 4.24. Then I calculate 70% from 4.24 and get 1.27, which will be a new distance between A…
Bob
  • 203
0
votes
1 answer

Rotating points with horizontal and vertical tilt

From Wikipedia, I've seen that if I have a rotation to do in three dimensions, it must be around an axis in order to do so. However, I have a rotation along the z-axis along with the xy-plane (aka a horizontal and vertical tilt respectively). How…
user192061
0
votes
1 answer

Position of a point on a line segent relative to the segent's length

I would like to ask for help with clarifying the following formula for calculation of relative position of a point on a line segment with respect to the line segment's length in two-dimensional Euclidean space: $$t = \frac{(P-A)(B-A)}{\lVert B-A…
Kyselejsyreček
  • 127
0
votes
3 answers

Can you construct (ruler and compass) a square with an irrational area?

I've heard that when $\pi$ was proved irrational, that squaring the circle was not proved impossible. This lead me to believe that you could construct a square with an irrational area. Is this possible? It is possible to create a polynomial with…
Gridley Quayle
  • 1,659
0
votes
1 answer

A level Ellipse question

An ellipse has the equation $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1$, where $a>b$, and with eccentricity $e$. It also has foci $S$ and $S'$ and directrices $l$ and $l'$. a) Use the focus-directrix property to show that $PS+PS'=2a$, where $P$ is a point…
Ahmed
  • 1
0
votes
1 answer

Prove that for any two points $A$ and $B$ $\overrightarrow{AB} \cup \overrightarrow{BA} = \overleftrightarrow{[AB]}$

Question: Prove that for any two points $A$ and $B$ $\overrightarrow{AB} \cup \overrightarrow{BA} = \overleftrightarrow{[AB]}$ The right hand side of the statement that I am trying to prove is a line AB in a set. It's just a single line AB in the…
usukidoll
  • 2,074
0
votes
1 answer

Orthogonal Coordinates

I'm hoping someone could give me a good definition of "orthogonal coordinates." Attempts to find one online has left me only with a vague idea. A reference text would be appreciated.
wellfedgremlin
  • 1,153
0
votes
0 answers

Mathematics-Oriented 4-D Glossary?

Is there somewhere a comprehensive glossary of words or phrases describing geometric concepts or objects in the Euclidean (not Einsteinian) fourth dimension? I have seen a glossary which purported to be such, but its terms seemed less like serious…
Senex Ægypti Parvi
  • 2,419
0
votes
1 answer

Tranversal parallel lines theorem

I need to prove Tranversal parallel lines theorem that says: If two parallel lines are cut by a transversal, the corresponding angles are congruent, the alternate angles are congruent, and the consecutive angles are supplementary. And other way…
user193541
  • 1
1 2 3
…
50 51