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

How can there be rotational degrees and more than one axis in space?

I apologize if this is a bit simplistic, but how can there be rotational degrees, or in simpler terms more than one "direction," even if we are speaking about a single plane? Is this just a supposition which is required in order for geometry to…
Anthony V
  • 1
0
votes
1 answer

Prove that: BD/DC = AR/AS

In a triangle ABC, internal bisector of angle A intersects side BC at D. R and S are circumcentres of triangle ABD and triangle ADC respectively. Then prove that, BD/DC = AR/AS.
RB MCPE
  • 26
  • 2
0
votes
2 answers

The size of the biggest square that can be inscribed within a circle

(a) Is the size the biggest square that can be fit inside a circle always $2r^2$? (b) How do you do when you show that is indeed the case? I want to compare my boyfriend's proofs with that of other people, since he thinks his proof is unique.
simulacra
  • 71
  • 1
  • 4
0
votes
0 answers

Is there a flaw in this proof?

A new user posted this question on Nov 16: "$MNPQ$ is a cyclic quadrilateral (that $M$ and $N$ are fixed points on the circumcircle of the cyclic quadrilateral $MNPQ$. The points $P$ and $Q$ can vary on the circle) with $K$ is the intersection…
Edward Porcella
  • 3,940
  • 2
  • 11
  • 15
0
votes
0 answers

How to express y in terms of x based on a diagram

The above diagram shows the intersection of two roads, their widths being 16m and 9m. If RN = x , and MP = y, how would you express y in terms of x. Is it by using the method of congruent triangles here? If so, how would it be done?
newbie2407
  • 1
0
votes
1 answer

How far can a triangle center be from the triangle?

As is well known, a triangle center can be exterior to the triangle. So, just how far from the triangle can one of its centers be?
user584285
0
votes
1 answer

pythagorean triples reverse calculation

what if i wanted to calculate m,n where C= 5 , for example $$ a = 2np; b= m2 -n2; c = m2 + n2; $$ let's say i want to start with c=5 instead of a m > n > 0 , how can i calculate m and n which created the values for a = 2mn , b= m2 -c2 , c = m2 +…
michael501
  • 149
0
votes
0 answers

Degrees of freedom in hyperplane intuition?

I can describe an $n-1$ dimensional hyperplane in $R^n$ with a point and a single $n$ dimensional vector (namely, the normal vector). Similarly, I can describe a $1$ dimensional hyperplane (a line) in $R^n$ with a point and a single vector (the…
Him
  • 447
0
votes
1 answer

prove that a point x is on the line perpendicular bisector of segment AB if and only if X.....

prove that a point x is on the line perpendicular bisector of segment AB if and only if X is the center of a circle through points A and B I'm unsure how to prove this. I will ask questions is the proof provided I cant understand thank you
Maximiliano
  • 1,121
0
votes
1 answer

Unique distance as a proof of a line and a plane parallelity

Please consider a statement "a line is characterized by a unique distance from a plane." Is the unique distance a sufficient feature to prove that the line and plane are parallel?
RobyK
  • 3
0
votes
0 answers

$P$ is a point inside the triangle $ABC$, from which perpendiculars intersect the sides $BC$ $AC$ $AB$ to the points...

$P$ is a point inside the triangle $ABC$, from which perpendiculars intersect the sides $BC$ $AC$ $AB$ to the points $D'$, $E'$, $Z'$. If $AD$, $BE$, $CZ$ are the heights prove that $$ \frac{PD'}{AD}+\frac{PE'}{BE}+\frac{PZ'}{CZ}=1.$$ I believe it…
zevs12
  • 1
0
votes
0 answers

Euclidean Geometry: Parallel Postulate and transitivity of parallelism

Definition: Two lines are parallel if they are coplanar and everywhere equidistant. Postulate 2: Through a point in a plane not on a line, one and only one line can be drawn parallel to that line. Are parallel lines equal to each other, but there's…
user593454
  • 1
0
votes
3 answers

Four points on the plane are vertices of three quadrilaterals. Explain how this happens.

This is a question from Kiselev's plane geometry book: Four points on the plane are vertices of three quadrilaterals. Explain how this happens. How do you explain this?
James Warthington
  • 3,187
0
votes
1 answer

Parallelogram in an isosceles triangle

Given an isosceles $ABC$ triangle ($AB=AC$), and points $D,E$ on sides $AC, AB$, respectively, such that $AD
user173628192
  • 1,506
0
votes
1 answer

In-center related. Prove that the indicated quadrilateral is also a rhombus.

AE and BD are the angle bisectors of $\triangle ABC$ with in-center I. P is any point on DE produced. X, Y, and Z are the feet of the perpendiculars from P to CB, BA, and AC respectively. (1) Through P, draw a line parallel to AB cutting BC at…
Mick
  • 17,141
1 2 3
…
50 51