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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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what is the maximum number of acute angles

What is the maximum number of acute angles in a convex 10-gon in the Euclidean plane ? I know that the answer is at least $4$. Any idea how to proceed.
user382915
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A hard problem from regular $n$-gon

I found the following theorem in a book of mine without a proof. Could someone show me a proof of it? Given a regular $n$-gon, with $n$ odd and vertices $v_1,\ldots,v_n$, and $C$ its circumcircle. At each $v_i$ draw a circle that is internally…
GeometryFan
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Why does the Pythagorean Theorem have its simple form only in Euclidean geometry?

Below are the right-angled forms of the Pythagorean Theorem in elliptic, Euclidean, and hyperbolic geometry, respectively. $$\cos\left(\frac{c}{R}\right) = \cos\left(\frac{a}{R}\right)\cos\left(\frac{b}{R}\right)$$ $$c^2 = a^2 + b^2$$ $$\cosh c =…
El'endia Starman
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Is triangle congruence SAS an axiom?

I was wondering if there is a way to prove SAS in triangle congruence with Euclidean axioms. Thank you for your help!
user157740
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The Pythagorean theorem and Hilbert axioms

Can one state and prove the Pythagorean theorem using Hilbert's axioms of geometry, without any reference to arithmetic? Edit: Here is a possible motivation for this question (and in particular for the "state" part of this question). It is known…
the L
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Does a maximal inscribed square in a regular polygon have a side parallel to a side of the polygon?

Suppose $P$ is a regular (i.e.,equiangular equilateral) polygon in the Euclidean plane, and the number sides of $P$ is not a multiple of $4$. Then $P$ contains an inscribed square. (Citation.) Of all the squares inscribed in $P$, some are largest.…
msh210
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Why does it make no sense to define addition on points in geometry?

Here is a simple question: Why does it not make sense to define addition on points in geometry? To be a bit more specific: assume we are talking only about standard Euclidean geometry. It is clear that a barycentric combination of points is…
Gab
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Prove that $XY$ crosses the midpoints

$\triangle ABC$ has altitudes $AD$, $BE$, $CF$. The reflections of $E$, $F$ in $H$ are $E'$, $F'$. The circle $DE'F'$ intersects $BE$, $CF$ at $X$, $Y$. Prove that $XY$ goes through the midpoints of $AB$, $AC$. I can show that $XY$ is parallel to…
Plato
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Can two planes intersect in a point?

Is it true that two planes may intersect in a point ? or If they intersect then, they always make a straight line ? I have some doubt; please explain.
ram
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Proving $R\ge 2 r$ using synthetic geometry

If $R$ and $r$ be the radii of the circumcircle and incircle of a triangle, then how do I prove by synthetic geometry(i.e. without trigonometry) that $R\ge 2r$? I am aware of a trigonometric proof but I am not quite sure if I can come up with a…
Richard Nash
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finding the center of a circle (elementary geometry)

I am considering the problem of finding the center of a given circle C of radius r. I currently know 3 elementary ways (ruler+compass construction) of doing this : - choose 2 points on the circle, draw the bisector, and draw another bisector…
Mathieu Dity
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Regular polygon diagonal lengths

Suppose that a regular $n$-gon has integer side length $m$. Is the lengths of its diagonals always algebraic numbers? If yes and if $n,m$ are given, is there an easy way to compute the diagonal lengths as a root of some polynomial?
selfstudying
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Prove three sides make a triangle from basic assumptions

I've been working through The Four Pillars of Geometry by John Stillwell. In exercise 2.5.3 he asks, How can we be sure that lengths $a,b,c>0$ with $a^2+b^2=c^2$ actually fit together to make a triangle? (Hint: Show that $a+b>c$) Lemma: $a+b>c$. …
ttt
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distance between any two interior points in a triangle less than largest side

How can one prove that the distance between any two interior points in a triangle less than largest side by elementary way using euclidian geometry. PS : I'm aware of this answer which cannot be delivered to school students Distance between interior…
ahmed
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Trilateration with unknown fixed points

I am able to measure my distance to a set of (about 6 or 7) fixed but unknown points from many positions. The difference in position between measurements is also unknown. I believe that I should be able to work out the relative position of the fixed…
FlightOfStairs
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