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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.

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a trapezoid, some similar triangles, and some homework

A high-schooler I know had this problem for homework: Given: trapezoid $ABCD$ with $\overline{AB}\Vert\overline{CD}$ and $m\angle C=90^\circ$; $E$ is the midpoint of $\overline{AD}$; $\overline{CB}$ is extended outside the trapezoid to $F$ such…
msh210
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Copying an angle

In The Non-Euclidean Revolution by Richard Trudeau, he discusses the theorems in Euclidean Geometry. In particular, I was struck by the proof of Theorem 23 (copying an angle). He avoids Euclid's proof because it used Theorem 8/SSS (which Euclid had…
genepeer
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Prove that MD=(d1+d2+d3)/3 if M is the centroid of the triangle ABC, and d1, d2, d3 and MD are perpendicular to the line under the triangle?

Here's a picture because I can't really explain the problem. Tried to prove it with using the fact that the centroid divides a median in a ratio $2:1$ and similar triangles but I got stuck.
pavle
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Prove that the distance of a centroid to a plane is the arithmetic mean of the distances of the vertexes of a triangle to that plane

I tried to use the fact that the centroid divides the median in a ratio of $2:1$ and using similar triangles, but I got stuck.
pavle
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Minimum number of vertices in a polygonal path

Let $\Omega\subset\mathbb{R}^n$ be a bounded connected open set. Given a positive number $R>0$ I want to find -if possible- the minimum number of vertices of a polygonal path joining two arbitrary points $x,y\in \Omega$ s.t. the length of the…
Oromis
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Elementary Geometry, proof:Isosceles triangle base \overline AB halves a length.

The assignment is as follows. Given the isosceles triangle $\triangle ABC$, on the line $AC$ a point is selected, lets call it $D$, also on the line $CB$ a point is selected, point $E$ such that $\left | AD \right |$= $\left | BE \right |$ AND…
DeLuini
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Proof of Euclidean Prop I.14 using Hilbert's plane axioms.

Rewrite the statement "If with any straight line, and at a point on it, two straight lines not lying on the same side make the sum of the adjacent angles equal to two right angles, then the two straight lines are in a straight line with one…
Faust
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In the following figure, $AD$ is the bisector of $\angle A$.Prove that: $\angle CBA = \angle DAB$

In the following figure, triangle $ABC$ is inscribed in circle $C$ and $AD$ is the bisector of $\angle A$.Also it's known that: $AD=BC$.Prove that: $\angle CBA = \angle DAB$ I tried as follows: It's obvious that $\angle DBC=\angle CAD$ , so…
Hamid Reza Ebrahimi
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Shortest path between two points that stays away from its rotated images

In the plane equipped with an orthonormal basis, let us consider the two points $A$ and $B$ whose coordinates are $(-2,0)$ and $(1,1)$, respectively. Is there a path from $A$ to $B$ (i.e. a continuous map $\gamma : [0,1] \to {\mathbb R}^2$, with…
Ewan Delanoy
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Is there a way to find the "center of mass" of an irregular polygon?

Just like the medicenter of a triangle is its "center of mass" if we cut it. What about some irregular shaped object? My first thought is to divide it up in triangles, then find the medicenter of each triangle, then connect the medicenters and…
pavle
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Regions of $4$-dimensional space

I am confused with the imagining the regions of $4$-dimensional space. I know that this space defines $2^4=16$ regions. Consider half upper space ($8$ regions). so $(2,3,4,1)$ belongs to the first region ($R_1$) and $(-2,3,4,1)\in R_2$. what we can…
C.F.G
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Find a radius of cyclic quadrilateral if given diagonals and an angle between them.

As I wrote, I have length of the diagonals $10$ and $6\sqrt{2}$ and an angle $45^{\circ}$ between them. What is a radius of this quadrilateral? I have no idea how to approach. I know there is partial answered…
nonuser
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What is the ratio of the area of the big circle to the sum of areas of the small circles if the triangle in the picture is equilateral.

I didn't really know how to translate the problem so i just drew a picture. I actually did this and I got 3 but when I checked again I got 4, and now I'm confused. Please let me know if there is something wrong with the problem and I'll clarify
pavle
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What is this geometric principle?

It appears that it does not matter how you move $A$, $B$, $C$, $D$, so long as $A$ and $D$ remain larger than $B$ and $C$, then you obtain the intersection between the three lines as shown ($y = x$, $[(A,B),(C,D)]$, $[(A,C),(B,D)]$).
Project Book
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Prove from Euclid's Postulates that C lies on line AB

Looking through the book "Euclidean and Non-Euclidean Geometries" by Marvin Jay Greenberg, there is the given problem: Given two points A and B and a third point C between them. (Recall that "between" is an undefined term.) Can you think of any way…
Jeffrey Walraven
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