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
2 answers

Geometry question solve without using trigonometrics

In $\triangle ABC$, $D$ is on $\overline{CA}$, $\angle B=120^\circ$, $\angle C=2x$, $\angle ADB=3x$, $\overline{AD}$=$\overline{BC}$ How do I find $x$? The solution is $x=10^\circ$, but I can't find it. I tried to rotate triangle BDC and make a…
sim
  • 13
  • 2
0
votes
1 answer

find distance given distances and right angles

In the following figure, it is given three right angles and distances : $ED = 3 $ , $ EB = 7$ and $CE = 5$. Is it possible to calculate the length $EA$. I tried using cyclic quadrilateral $ABCD$ and angles but couldn't find the expression. thanks…
ahmed
  • 1,273
  • 6
  • 10
0
votes
0 answers

Applying Playfair's axiom when the point lies on the given line

Playfair's axiom states: Through any point in the plane, there is at most one straight line parallel to a given straight line. This axiom is equivalent to the parallel postulate. If the point $P$ lies on the given straight line $\mathcal{l}$,…
Siddhartha
  • 1,770
  • 18
  • 38
0
votes
0 answers

Two lines in space must intersect to be perpendicular between them?

Or is not necessary and why? Where is the definition?
Carlitos_30
  • 209
0
votes
0 answers

Distinct lines with a common segment: why is this not ruled out by Euclid's postulate 1?

Euclid's first postulate is to draw a straight line from any point to any point, and in the Elements this is taken as including uniqueness. In a commentary on Euclid by David Joyce, I recently came across the following complaint about…
user13618
0
votes
0 answers

Postulate V implies equidistant postulate

(A) : The set of points in one side equidistant from a given line forms a line Postulate V : If two lines are drawn which intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines…
HK Lee
  • 19,964
0
votes
0 answers

Can a quadrilateral ABCD of area 1 be such that all points P have the property that at least one triangle (PAB, PBC, PCD, PDA) has an irrational area?

Prove or disprove that there exists a quadrilateral ABCD of area 1 such that all points P have the property that at least one triangle -- PAB, PBC, PCD, PDA -- has an irrational area. I have a feeling that no ABCD exists. My method is trying to…
Baker5680
  • 451
0
votes
1 answer

Prove that the points $Α,Β,Δ$ and $Ε$ are homocyclic

$ΑΒΓ$ triangle with $ΑΔ⊥ΒΓ$ and $ΒΕ⊥ΑΓ.$ Prove that (a) The points $Α,Β,Δ$ and $Ε$ are homocyclic. (b) $ΔΕ∥(ε)$ where $(ε)$ is the tangent at $Γ$ of the surrounding circle of the triangle $ΑΒΓ$.
user709021
0
votes
3 answers

Show that the Points $E,H,Δ,Ζ$ belong to a circle

Show that the Points $E,H,Δ,Ζ$ are homocyclic
user709021
0
votes
1 answer

Why weights are not squared in the weighted Euclidean distance formula?

I am working on Data Mining and need distance between data points. With two points $(x_1, y_1), (x_2, y_2)$ & weights $w_1$ and $w_2$ respectively. The euclidean distance is: $$ \sqrt{w_1\times(x_2-x_1)^2+w_2\times(y_2-y_1)^2}. $$ I think…
mrityunjoynath
  • 13
0
votes
1 answer

Intersection between two finite planes

I have two planes defined by three points each. These planes are "finite", meaning that the three points define their limits. These planes may or may not intersect, if so, the intersection is a finite line. What's the smartest way to find the two…
Rafael March
  • 13
0
votes
1 answer

Relationship between the number of variables in the equation and the number of dimensions of its graph

The graph of the equation $x = 0$ has one dimension, as it is a vertical line. The graph of the equation $x + 2y = 0$ has 2 dimensions, because it is just a negatively sloped line. The equation $x + 2y + 3z = 0$ is proven to be a plane (3…
Andrew
  • 3
0
votes
0 answers

Adrian-Marie’s Theory of Numbers Book

Recently I have found out about this famous book from the french mathematician Adrian-Marie called the theory of numbers, I think citing eculidians elements. I really want to read this book but don’t know where to find it since any copy on the…
Ned
  • 21
0
votes
1 answer

Slicing a circle's surface area in 3 equal parts not the usual way

Is there a method to slice a circle's surface area in three equal parts by slicing the circle using two straight lines whose common point of origin is located on the circumference of the circle?
Niko
  • 1
  • 1
0
votes
1 answer

Quadrilaterals with circles generated by internal bisectors

Prove that intersections of internal bisectors of a quadrilateral form a cyclic quadrilateral.
Piquito
  • 29,594
1 2 3
…
50 51