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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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The role of string in constructive geometry

I was wondering whether, if I add string and thumbtacks to my geometry kit, I am able to do any new constructions. The idea being, with string, I can draw ellipses for instance and the intersection of these with lines, circles and other ellipses…
Mike F
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Pythagoras vs the manhattan distance

Suppose I write down the Manhattan distance from the origin to a point (x,y) in terms of a series of n steps of length x/n in the x direction, alternated with m steps of length y/m in the y direction: $$d_{Manhattan} = \sum_{i=1}^n \frac{x}{n} +…
Robert Sim
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What's this math concept visualized in this math poster including Euclid?

What's this math concept visualized in this math poster including Euclid? It looks similar to what I've seen in algebra regarding groups and their visualization, but aren't groups much more modern concept?
mavavilj
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Euclidean Geometry Problem

"Let $k$ and $l$ be two circles that intersect in two points $P$ and $Q$. Construct (with a straightedge and a compass) the line $m$ through $P$, not containing $Q$, with the property that if $m$ intersects $k$ in $B$ and $P$, and $m$ intersects $l$…
User New
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Euclidean Version of Pappus's theorem

I'm going through Hartshorne's Geometry, and one of the exercises has stumped me for a good few hours. The problem is a version of one of Pappus's theorems: Let $A$, $B$, $C$, be points on a line $l$, and let $A'$, $B'$, $C'$ be points on a line…
yunone
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Proving an exercise from my High School Geometry Class

In my class we are learning geometry and the instructor gave us this problem: Let $ABC$ be a scalene triangle with orthocenter $H$, and let $W$ be a point on the side $BC$, lying strictly between $B$ and $C$. The points $M$ and $N$ are the feet of…
Alan Yan
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Minimum Perimeter of a triangle

I have been playing the app Euclidea, I have been doing quite well but this one has me stumped. "Construct a triangle whose perimeter is the minimum possible whose vertices lie on two side of the angle and the third vertex is at point a" see image…
Mike
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Is Euclidean Geometry studied at all?

Is there a place for Euclidean geometry in the hearts or minds of any mathematicians? I personally find it to be the most beautiful mathematics I have yet encountered but I see little of it on sites such as these, leading me to believe that it…
Faraz Masroor
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Three mutually-tangent circles have centers at given distances from each other; find each radius, and find the area between the circles

Three circles of different radii are tangent to each other externally. The distance between their centers are $9\ cm$, $8\ cm$, and $11\ cm$. Find the radius of each circle. Find the area in between the three circles. The distance between the…
richmond
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Proving an angle to be 90

$H$ is the orthocenter of acute triangle $A B C$. Let $\omega$ be the circumcircle of $B H C$ with center $O^{\prime}. \Omega$ is the nine-point circle of $A B C . X$ is an arbitrary point on arc $B H C$ of $\omega$ and $A X$ intersects $\Omega$ at…
user1055487
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Prove converse Thales theorem, proportional sides imply parallel lines

I'm going through John Stillwell's Four Pillar's of Geometry and trying to follow the book's structure when doing the exercises. Generally, a 'pillar' is divided into two chapters; the first chapter states useful results and motivation while the…
ttt
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Definition of a right angle

I don't understand what a right angle is. Of course, I know what a right angle is, but I feel I don't understand it. I'm looking at Euclidean geometry of the plane. When looking at it from analytic geometry, everything is fine, but there the…
Gyro Gearloose
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euclidean geometric construction

this is question 42 in the red book of mathematical problems by k. s. williams and k. hardy. let abcd be a convex quadrilateral. let p be the point outside abcd such that $|ap| = |pb|$ and $\angle apb = 90^\circ.$ the points $q, r, s$ are…
abel
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Look for a clever geometric approach to solve a problem with area.

In the acute $\triangle ABC$, the position of each point is as shown, $F$ is the midpoint of $BD$, $G$ is the midpoint of $CE$, $F$ is not on $CE$, and $G$ is not on $BD$. It is known that $S_{\triangle AFG} = 1$, and the area of ​​the…
Zuo
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