Mathematics Stack Exchange
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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.

9328 questions
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Given two lines, each defined using Hesse normal form find the intersection point.

A line can be defined by two numbers : distance of line to origin and the angle from the origin to closest point on the line. Is there a formula to find the coordinates of intersection of two lines given this representation? I can go to y=a*x+b…
Stepan
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Does the set of distances characterises class of a finite set of points in the plane up to isometry?

Given a finite set of $n$ points on the plane. Is it fully characterized up to translations/rotations/reflections (isometric transformations) by the (multi)set of its $(n-1)n/2$ distances between pairs of distinct points? Example: If we consider the…
Samuel Vidal
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In the following figure prove that $AE||EF$ (without using similar triangles)...

From an arbitrary point $A$ outside a circle draw a tangent $AB$ to the circle and select point $E$ also outside the circle such that $AB=AE$.Let $D$ be an arbitrary point on perimeter of circle , call the other intersection of $AD$ with circle as…
Hamid Reza Ebrahimi
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Another formula for the angle bisector in a triangle

I have seen in an old geometry textbook that the formula for the length of the angle bisector at $A$ in $\triangle\mathit{ABC}$ is \begin{equation*} m_{a} = \sqrt{bc \left[1 - \left(\frac{a}{b + c}\right)^{2}\right]} , \end{equation*} and I have…
user232552
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Is there a theorem of intersecting chords in an ellipse?

I found a well known theorem that if $A,B, C$ and $D$ are on the circumference of a circle and $AB\cap CD=P$ then $AP\cdot BP=CP\cdot DP$ . Is there anything generalization of it to an ellipse? Maybe something that in a given ellipse, if $P$ divides…
curious
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Ratio of sides in a triangle vs ratio of angles?

Given a triangle with the ratio of sides being $X: Y : Z$, is it true that the ratio of angles is also $X: Y: Z$? Could I see a proof of this? Thanks
JohnDoe
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Average distance between nearest neighbors for randomly placed points in a unit square?

The answers I found were generally about the distance between any two points in a square. I'm trying to find the average distance between nearest neighbors. Background on this is I'm processing 3D point cloud data. Generally the points define a…
James C
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Proof that bisecting a line segment with straightedge is impossible

This is the proof I read from here. I will quote it fully: The answer is NO. To see why, consider a line L in the plane P, and two marked points A, B on it. It is desired to construct the midpoint M of the segment AB using the straightedge.…
vladimirm
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Find area of rhombus

Given the following rhombus, where points E and F divide the sides CD and BC respectively, AF = 13 and EF = 10 I think the length of the diagonal BD is two times EF = 20, but i got stuck from there.
sishke
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What happens to the angles of an isosceles triangle if one vertex is at infinity?

My son and I were trying to decide whether an isosceles triangle can ever have 90 degree base angles. I would argue that if the two equal length sides are both infinitely long, they must have 90 degree angles, because any angle less than 90 degrees…
Jen
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circumscribing sphere of tetrahedron

What are the conditions under which the center of circumscribing sphere of a tetrahedron is located inside(outside, face, edge) of the tetrahedron? In other words, how can we define acute(obtuse) tetrahedron?
Chung. J
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Calculate coordinates on a line (or ray)

If I have a ray, that starts at the origin, and has a slope of 0.5, how would I calculate the coordinates of a point at length 3 away from the origin (that's on the ray)? This isn't homework; I learned this a long time ago, but now it's gone, and I…
Adam Rackis
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Stupidly simple geometry problem I can't do

Okay. Here it goes. C and D are two points on the same side of a straight line AB and P is any point on AB. Show that PC + PD is least when the angles CPA and DPB are equal. I have no idea why I can't do this. I've drawn diagrams, tried using the…
Soham Chowdhury
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Constructing two tangents to the given circle from the point A not on it

I'm trying to complete Level 21 from euclid the game: http://euclidthegame.com/Level21/ The goal is to construct two tangents to the given circle from the point A not on it. So far I've figured that the segments from B to the tangent points must be…
Pjotr
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Simple 9-th grade geometry problem

I have a geometry problem which states that Find the range of $x$ in following figure. Given that $AD$ and $AC$ are equal, and the values and angles are also given. How to estimate the range of the problem possibly using basic geometry methods?…
Mula Ko Saag
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