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
1 answer

Proving elementary property about hyperplanes.

I am currently working through a textbook, and I am having some problem with the following problem: Define a hyperplane to be an $(n-1)$-plane of $E^n$. Prove that $P$ is a hyperplane if and only if $$P = \{x \in E^n \: | \: a \cdot (x-x_0) = 0…
Andrew Thompson
  • 4,431
1
vote
1 answer

Which of these are constructible numbers?

Which of these are constructible numbers? $$-\frac35\quad,\quad27^\frac16+2i\quad,\quad2^\frac13\quad,\quad e^{\frac{\pi i}{10}}$$ Please tell me how you came to the answer too. Thanks!
Eric Kim
  • 169
1
vote
1 answer

Find the area of the region formed at the intersection of all these arcs (GEFH) in terms of a.

ABCD is a square and the arcs centered at the vertices of the square and the radii, are all equal to the side-lengths of the square, (=a). I feel like the arc lengths should all measure 90 degrees as they are the arcs subtended by the angle at…
user146026
  • 67
1
vote
1 answer

Find the measure of the angle ADE and the radius of the circles in terms of the sides of the square ABCD.

We know that the triangles are congruent. The circles are congruent. And EFGH is a square. I don't even know where to begin with this one besides to say that angle ADE is equal to 90-m(EAD), but I feel like it's probably asking for something more…
user146026
  • 67
1
vote
1 answer

Specific triangle, symmetral of angle proof

In triangle $ABC$, angle $\gamma = 120$. Prove that $|\overline{CC'}|=\tfrac{ab}{a+b}$, where $\overline{CC'}$ is symmetral of angle $\gamma$ inside triangle. Look at image. I can't use areas, becuase we haven't learned them yet. Our teacher says…
user195929
  • 11
1
vote
3 answers

Has triangle an angle?

I read axiomatic geometry and found the following definitions: Points $A$ and $B$ and all those points that lie between those points is a line segment. If $AB$ and $AC$ are two rays that does not belong to the same line, then the union of those rays…
Idon'tknow
  • 13
0
votes
2 answers

Constructing a triangle given two angles and the sum of two sides

I am looking for an answer to the following construction Construct a triangle given two angles (3 angles) and the sum of two sides
Philippe Sussholz
  • 187
0
votes
1 answer

What goes the Pappus' theorem says

I found the following statement: Let $A, C$ and $E$ be three distinct points on the line $l_1$ and $B,D,F$ three distinct points on the line $l_2$. Let us assume that $AB\cap DE=L$, $CD\cap FA=M$, and $EF\cap BC=N$. Then $L,M,N$ are collinear. In…
guest
  • 1
  • 1
0
votes
1 answer

How to prove the Pitot's theorem?

I read the following Pitot's theorem: A quadrilateral ABCD is tangential if and only if $AB+CD=AD+BC$, where $AB$ means the length of side $AB$. How can I prove it. I mean, the case $ABCD$ is tangential $\implies AB+CD=AD+BC$ is easy because there…
geometrystudent
  • 47
0
votes
1 answer

Measure of angles is the same

I found the following theorem in a lecture notes without proof: Let $A, B, C, D, E$ and $F$ be points on the plane such that $\angle ABC$ and $\angle DEF$ are either both acute or they are both obtuse. If $AB$ is perpendicular to $DE$ and $BC$ is…
geometrystudent
  • 47
0
votes
1 answer

How to prove the tangent secant theorem

I was reading the the following theorem: Let $A,B$ be two points on the circumference of a circle. Let $C$ be a point outside the circle. Then $\angle BAC=\frac{1}{2}\widehat{AB}$. Is there some elementary way to prove it? Here is a picture with…
geometrystudent
  • 1
0
votes
0 answers

Synthetic proof of curvature formula

The radius of curvature of a curve $\gamma:I \to \mathbf{R}$ parametric by arc length is $||\ddot \gamma||^{-1}$. I want to demonstrate this using synthetic geometry. Let $A$, $B$ and $C$ be three noncolinear points with $AB = BC = 1$ and let $O$…
michaelkovarik
  • 431
0
votes
2 answers

Rational distance implies countable set

I am working on this problem for weeks without a good solution. Let $S\subset\Bbb R^d$ be a set in which $\rho(s_1,s_2)\in\Bbb Q$ for any $s_1,s_2\in S$, where $\rho$ is the Euclidean distance in $\Bbb R^d$. Show that $S$ is countable. If…
YYF
  • 2,917
0
votes
0 answers

Prove Proposition I.7 of Euclids for the case where D lies in the interior of triangle ABC

Prove that Given triangle ABC with apex C, we cannot construct another triangle ABD with D lying in the interior of ABC Is this proven the same way as if D lies on the same side of AB? I have it started as: Given Triangle ABC we want to show that no…
Allison
  • 41
0
votes
1 answer

Is there a name for this point?

I found the following problem interesting: In a three villages $A$, $B$ and $C$ there are $a,b$ and $c$ pupils respectively. Where should one build the school such that the total length of pupils going to the school is as small as possible? How one…
curious
  • 1
1 2 3
…
50 51