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
2
votes
3 answers

intersection between a line and a sphere

I have a plane sphere inscribed in a cube like in the image below. Both the sphere and the the cube are centered at the origin. The cube's edge has a unit length (so one edge of the cube is (0.5,0.5,0.5) Take a point A(x,y,z) situated on one of the…
Ryan
  • 309
2
votes
2 answers

Euclid the Game Level 2: Construct a line that bisects the given angle.

This is the level I'm trying to complete: My last idea was this, but I still don't get a message that I completed the level. Why is this not correct ? Last time I did mathematics was 10 years ago, sorry if this is a real noob question, but I…
Pjotr
  • 309
  • 1
  • 4
  • 8
2
votes
1 answer

How to prove $C=2πr$?

Everybody know this formula,but why the relation between $C$ and $r$ is linear relation? Not $C=2πr^{0.99}$ or $C=2πr^{1.01}$how to prove it,what axiom is it based on?
Tom
  • 671
2
votes
1 answer

About a solid which satisfies $\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n)$

For $n\ge 2\in\mathbb N$, let $S_n$ be the volume of a $(n-1)$ dimensional solid which satisfies $$\sum_{i=1}^{n}x_i=0, |x_i|\le1\ (i=1,2,\cdots,n).$$ Then, here is my question. Question : Can we represent $S_n$ by $n$ ? Motivation : I've been…
mathlove
  • 139,939
2
votes
2 answers

About the inscribed sphere and the exspheres of a $n$-dimensional simplex

Let us consider $n$-dimensional simplex $K$ in $n$-dimensional Euclidean space. Let $r_0$ be the radius of the inscribed sphere of $K$, and let be $r_1, r_2, \cdots, r_{n+1}$ be each radius of the exsphere of $K$. Then, here is my question. Question…
mathlove
  • 139,939
2
votes
1 answer

How is bisector of one side of a right angled triangle, drawn from right angled corner equal to the half of the bisected side?

In a right angled triangle ABC with right angle at B and D being the mid-point of side AC, is it possible to prove BD=AD=CD without using co-ordinate geometry or circle theorems etc? (Just by using other elementary theorems: This has bothered me…
user1077777777777777
  • 21
2
votes
2 answers

Length of Chord is Independent of Point P

This is question 1.49 from Baragar's textbook called A Survey of Classical and Modern Geometries if anybody is familiar with that text. This question is assigned as homework, so I am just looking for some places to start, not the full answer. The…
ruferd
  • 534
2
votes
3 answers

perimeter of square inscribed in the triagle

In the figure given below, PQR is a triangle with sides PQ=10, PR=17, QR=21. ABCD is a square inscribed in the triangle. I want to find perimeter of square ABCD that is to find the length of side AB. But by using of basic high school geometry…
curious_mind
  • 1,434
2
votes
0 answers

Given two triangles, if the vertices of the first triangle coincide with the vertices of the second triangle, are the triangles are equal?

I'm going through Euclid's Elements (Book 1) (Proposition 4). We're given two triangles (ABC, DEF) where two sides of the first triangle are equal to the two other sides (say AB=DE and AC=DF) of the second triangle. The angle between these two sides…
nemo
  • 63
2
votes
1 answer

angle bisector and relationships between lengths

in triangle $ABC$, bisector of ∠A and segment $BC$ meets in $D$. $AD+AC=BC$, and $AD+AB=CD$. if we let ∠C=x°, how much is x? approach i tried to use $AB:AC=BD:DC$, but it was hard to find the solution.
Chokbitt
  • 47
2
votes
1 answer

Continuous Functions Defined on Spheres

There is Dyson's theorem that states that every continuous real-valued function on the 2-sphere there are four points $p_i$ that form a square around the origin of the sphere and $f(p_1)=f(p_2)=f(p_3)=f(p_4)$. see:…
user1105749
  • 31
2
votes
0 answers

Shortest halving line

Elsewhere I read a proof that a diameter is the shortest curve bisecting a circle (intuitively obvious), but an altitude is not the shortest bisector of an isosceles triangle (not obvious). Is anything known about the shortest bisector of an…
Hauke Reddmann
  • 735
2
votes
1 answer

Help on whether a geometry solution is valid.

Let $ABC$ and $AB'C'$ be similar right angled triangles with right angles at $C$ and $C'$, respectively. Let $l$ be the line between $C$ and $C'$, and let $D$ and $D'$ be the points on $l$ such that $BD$ and $B'D'$ are perpendicular to $l$. Prove…
John Marty
  • 3,650
2
votes
2 answers

Euclidean Geometric Problem: prove that line segment ratio is $1:2$

I'm working on this plane geometry problem: $ \bigtriangleup AEF$ is a triangle, $\angle A = 60^\circ$, $\angle F = 40^\circ$, $\angle E = 80^\circ$, points $K$ and $C$ are on line $AF$, $\angle CEF=20^\circ$, $\angle KEF=40^\circ$, $O$ is the…
Xiaoyong Guo
  • 151
1 2 3
…
50 51