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
1
vote
2 answers

$AB=AC$, $BD$ is the angle bisector of $\angle B$ , find $\angle A$

Let $ABC$ be an triangle, $AB=AC$. $BD$ is the angle bisector of $\angle B$, $BD$ intersect $AC$ at point $D$, and $AD=BC+BD$. show that: $\measuredangle BAC=20^\circ$ Well, If $\measuredangle BAC=20^\circ$, I can prove that $AD=BC+BD$. But, if…
ziang chen
  • 7,771
1
vote
0 answers

what is exactly the fine-scale geometry of Euclidean space?

I have learnt math for a while, but I really do not have impressions on the concept of fine-scale geometry of Euclidean space. I hope this is not a trivial question as it sounds like. Any comments would be greatly appreciated.
nstrong
  • 163
1
vote
0 answers

How can I show that symmetry groups of two regular n-gons are conjugate?

Let $P_1$ and $P_2$ be regular n-gons in $E^2$ with centers $C_1$ and $C_2$. Prove that $Sym(P_1)$ and $Sym(P_2)$ are conjugate in $Isom(E^2)$. Hi, I have to prove this but it looks very obvious to me. But I don't know how to write it and where can…
floran
  • 11
  • 1
1
vote
3 answers

Two prove two lines in a triangle are parallel

$D$, $E$, $F$ are the midpoints of sides $BC$, $CA$ and $AB$ respectively of a triangle $ABC$ right angled at $C$. If $EF$ and $DF$ (extended if necessary), meet the perpendicular from $C$ on $AB$ in points $G$ and $H$ respectively, show that $AG$…
lokesh sangabattula
  • 657
1
vote
3 answers

Reflections in Euclidean plane

Let $T: \mathbb{R}^2 \to \mathbb{R}^2$ be the counterclockwise rotation of $\frac{\pi}{2}$ and $S: \mathbb{R}^2 \to \mathbb{R}^2$ be the reflection w.r.t. the line $x+3y=0$. There exists a reflection $R$ such that $T^{-1}ST=R$? Is there a canonical…
TheWanderer
  • 5,126
1
vote
1 answer

Proof that an angle across line is equal to 180 degrees

When given a straight line, how do you prove that an angle across it is equal to 180 degrees, or two right angles? It feels like something that should be an axiom, but it isn't one of the 5 postulates of Euclidean geometry. I suspect the proof is…
Cataline
  • 2,018
1
vote
1 answer

Prove that angles in a circle equal each other

Let a circle in the Euclidean plane be given, let $AB$ be a diameter, and let $CD$ be the tangent through point $A$. Let $E$ and $F$ be two points on the circle, on the same side of $AB$ as $C$, and with $F$ between $E$ and $A$. Show that…
Elizabeth Hill
  • 37
1
vote
0 answers

Finding the unit quaternion for the regular octahedron

I am given this figure of a regular octahedron and I have to find the unit quaternion. In the symmetry group of the regular octahedron what is the unit quaternion that represents a 90 degree rotation about the semi x-axis $\lambda i, \lambda >0$ .…
usukidoll
  • 2,074
1
vote
1 answer

Prove that $GEBD$ is a square (see diagram).

$ABB_1A_1,BB_2C_1C,ACC_2A_2$ are squares. The problem itself is to prove that the area of $ABC$ and the area of $BB_1B_2,CC_1C_2,AA_1A_2$ are equal. If I could only prove that GEDB is a square it would follow that $\triangle AEB \cong \triangle…
alexgiorev
  • 1,262
1
vote
2 answers

Equilateral Triangle equality

Let ABC be an equilateral triangle, and P be an arbitrary point within the triangle. Perpendiculars PD, PE, PF are drawn to the three sides of the triangle. Show that, no matter where P is chosen, PD + PE + PF / AB + BC + CA = 1/2√3
Pita Frita
  • 13
1
vote
2 answers

In triangle ABC, ∠ACB = 60. AD and BE are angle bisectors. Prove AE + BD = AB.

Here is a diagram if it will be helpful:
alexgiorev
  • 1,262
1
vote
1 answer

Travelling Sales Man problem

I am studying the Travelling Sales Man problem: Given a list of cities and the distances between each pair of cities, what is the shortest possible route that visits each city exactly once and returns to the origin city? And I wonder: given a…
Maestro
  • 165
1
vote
1 answer

Geometric meaning of the sum/product of slopes

Consider two lines in the plane $\mathbb{R}^2$ of slopes $s_1$ and $s_2$. Is there a geometric meaning of the sum $s_1+s_2$, difference $s_1-s_2$, product $s_1s_2$ or quotient $s_1/s_2$ of the slopes ? Put differently, is there a geometric…
Fred
  • 77
1
vote
1 answer

Triangles formed by diagonals of trapezoids

$\Delta$ AOB and $\Delta$ DOC should be equal in area. Correct me if I am wrong. Given: Trapezoid ABCD with ratio $\frac{area \Delta AOB}{area\Delta ABD}$ = $\frac{3}{4}$. I am trying to find (1) Ratio of Area $\Delta$AOD to $\Delta$BOC; (2) Ratio…
Rex
  • 249
1
vote
3 answers

Is there an isometry $\mathbb R^3\to \mathbb R^3 $ with this property?

What is an isometry $\mathbb R^3\to \mathbb R^3 $ which maps $(3,1,2) \mapsto (2,2,2)$ and $(2,2,2) \mapsto (1,1,2)$?
Madhav Bapat
  • 341
1 2 3
…
50 51