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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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How to fit a convex quadrilateral inside another short of cutting them out and playing with them?

I have two convex quadrilaterals (ABCD and WXYZ). Their sides and their interior angles are known. I also know that WXYZ fits perfectly inside ABCD with each corner point touching a different side. Is there any way to figure out analytically…
SJ.Storms
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Prove perpendicular bisectors of non-parallel lines intersect

Suppose that $A$, $B$ and $C$ are points and that $AB$ and $BC$ are not parallel. Show that the perpendicular bisector of $AB$, $l$, and the perpendicular bisector of $BC$, $l'$, are not parallel and so intersect.
AndroidFish
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Find distance of overlapping squares

How to find distance center to center from square $1$ to square $3$, if we need overlap area is $15.46 mm^2$. if we know each side of the square is $6.9 mm$. Firstly I find the distance is $9.3 mm$ but it's wrong. Anyone can help to solve this…
Roeny
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Ambiguous case in euclidian geometry

Is it possible to locate $4$(or more than $4$) points on a plane such that every point is at an equal distance from every other point?
Ashwin Pandey
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Prove transformation isn't a translation

I have these maps from the plane to itself where $X=(x,y)$: $f(X):=(y,-x)$ $g(X):=(x+2y,y)$ I need to compute $fg$ and $gf$ and show that none of these compositions are simply translations or that for a point $(x,y)$, it shouldn't be mapped to…
User123454321
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Construct an equilateral triangle w. vertices at given distances from a point

I am self-studying Euclidean Geometry. One problem with studying alone is that when you’re stuck, you’re stuck. I can’t get past the following problem: “To describe an equilateral triangle having given the distances of a point from each of its…
BlakeDavis
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Do Euclidean geometry preservers parallelism of lines and area ratios?

Do the Euclidean geometry preserves the properties parallelism of lines and area ratios for any possible transformation? I know that the Affine geometry do and I think that Euclidean geometry also do it.
kotakata
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An object is placed in front of a plane mirror of length $L$ ...

I am stuck on the following problem : An object is placed in front of a plane mirror of length $L$ at a distance $d$ of its bisector line .An observer is at a perpendicular distance of $2d$ from the mirror.If the observer is walking parallel to the…
learner
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Euclidean Geometry

$XYZ$ is a triangle in which $\angle X$ is obtuse. A point $P$ is taken inside the triangle and $XP$, $YP$, $ZP$ are produced to meet the sides $YZ$, $ZX$, $XY$ at the points $K$, $L$, $M$, respectively. Suppose that $PL = PM$. Find the angles of…
Lama
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Prove the following statement

Prove that the point $P=(x',y')$ will be inside the acute or obtuse angle made by $a_1x+b_1y+c_1= 0$ and $a_2x+b_2y+c_2= 0$, if $$(a_1x'+b_1y'+c_1)(a_2x'+b_2y'+c_2)(a_1a_2+b_1b_2)\quad <\text{ or }>\quad0$$
Symon Saroar
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Prove that there exists an isometry P in Isom(H^2) such that P(A)=B

Where A and B are ideal triangles in H^2 (upper sheet of hyperboloid). How do I get started with this proof?
John.P
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Shortest expression for the diagonal in a convex non-cyclic quadrilateral knowing its sides and the other diagonal?

I'm trying to arrive to the shortest expression possible for finding the diagonal in a convex (and non-cyclic) quadrilateral, knowing its four sides lengths and the other diagonal. My best try matches the second proposed solution in this other…
cesss
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Geometry with circles.

Two circles, with centres O and P respectively, intersect at A and B. The extension of OB intersects the second sircle at C and the extension of PB intersects the first circle at D. A line through B parallel to CD intersects the first circle at Q…
user291241
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A problem on Euclidean Geometry

How can i prove that if a triangle has sides of lengths a, b, e, then its area S satisfies the inequality $$4\sqrt{3}\leq a^{2}+b^{2}+ c^{2}$$ with equality holding only for equilateral triangles. (Hint: If $\theta$ is the angle between sides $b$…
user
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An exersise of Euclidean geometry

The following question is one of the exercises "Foundation Euclidean and non-Euclidean geometry" by Greenberg (chapter 1/ Major Exercises/ 3 ) For any angle, draw a circle $\gamma$ radius $d$ centered at the vertex $O$ of the angle. This circle cuts…
user
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