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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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Seeking for construction s.t. every intersection contains at least 3 lines

In Euclidean geometry, is there some set of lines in s.t. there are at least 2 intersections, but every intersection contains at least 3 lines, and no lines in the set are parallel? I tried for a long time to construct this by hand but couldn't find…
simonzack
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how many circle of radius r can be placed inside on the border of another circle

Suppose Radius R of a big circle is given. and I want to place some little circles of radius r inside on the border of that big circle. Like the picture: But how to find how many small circle can be placed on the border?
HammrerEngineer
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Obvious statement, but how do you formally prove it?

Let d be a line, M a point on the line, and n a positive integer. Why is there exactly two points at n distance from M on d? How to prove it with Euclidian axioms (without algebra) ?
Jor
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Euclidean geometry of the ratio of two lengths

Let ABC be a triangle with $AB=BC$ and $\angle{ABC}=90^{\circ}$. Let $D$ be the midpoint of $AC$, and $E$ be a point on the opposite of $AC$ as $B$ and $\angle{AEC}=45^{\circ}$. Is the ratio of $EB/ED$ constant? If so, what is its value?
Qiang Li
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Geometry problem: Find the ratio of side of parallelogram to produced side.

I found this geometry question in IMO-2015 (conducted by the Science Olympiad Foundation, India) paper. It goes like this: ABCD is a parallelogram and L is a point on DB. The produced line AL meets BC at M. Given that DL = 3LB, find $\frac{AB}{CN}$…
Arya
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Distance between two points in distinct circles

Suppose we have two circles with radii $\delta>0$ and the distance between circle is $r$, i.e. $|AB|=r$. Let $x$ and $y$ be points in distinct circles. How to prove rigorously that $|x-y|>r$? I've tried to use triangle inequality but it did not work…
RFZ
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Proof of Construction of the symmedian

I read this proof about the construction of the symmedian. It says that if we want to construct the symmedian for a triangle $ABC$ one way is to let the intersection of the tangents to points $B$ and $C$ be $D$. Then $AD$ is the symmedian. Here is…
PNT
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David Hilbert's Foundations of Geometry, section 9, Compatibility of the Axioms

I am reading David Hilbert's Foundations of Geometry. In section 9, where he shows the "Compatibility of the Axioms" he begins with the following: Let us consider the domain $\Omega$ consisting of all those algebraic numbers which may be obtained…
user593257
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Do we need measure theory to prove length of diameter and circumference is constant is a circle?

In high school I was taught that $\pi$ is the ratio of the length of circumference and diagonal of a circle. But is it necessary to use some measure theory machinery to define the length of circumference of a circle? Or length of a diameter? And…
curiousamateur
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Find ratio of segments in triangle

We are given a triangle $ABC$ and points $D$ and $E$ on $AB$, such that $AE = ED = DB$. $F$ and $G$ are arbitrary points on $AC$ and $BC$ respectively. We want to find the product of ratios $\frac {EH}{HF} \cdot \frac{FM}{MB}$. I have tried drawing…
Pradeep Suny
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Distance between two axis-parallel ellipses

I have two axis parallel ellipses. I want to find the gap (minimum distance) between the two. The ellipses are defined as : $$\frac{(x-x_1)^2}{a_1^2} + \frac{(y-y_1)^2}{b_1^2} = 1$$ $$\frac{(x-x_2)^2}{a_2^2} + \frac{(y-y_2)^2}{b_2^2} = 1$$ I am not…
Suresh
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Diagonal triangle is autopolar

I was searching things about the diagonal triangle of a quadrilateral. I found in MathWorld's "Diagonal Triangle" entry the following question: If the quadrangle is a cyclic quadrilateral, then the circle is the polar circle of the diagonal…
Nana
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How can I prove that the cyan triangle in the picture is isosceles?

There is a circumference $\gamma$ which has a square $ABCD$ inscribed. There is also a semi-circumference $\gamma_1$ centered in $H$ with radius $\overline{OH}=\overline{AH}=\overline{HB}$. Then there is a half line having origin in $A$ and…
Damiano Scevola
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How to find area covered by a car windshield wiper when it swaps a certain angle?

The problem is as follows: A buggy is set to cross over a snowy terrain. The driver seat has a rectangular window featured in the diagram from below. When the driver activates the mechanism for cleaning the window from the snow, the wiper spins…
Chris Steinbeck Bell
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Prove that $AI=PI$

Given a triangle $ABC$ with incenter $I$, line $AI$ intersects the circumcircle of triangle $ABC$ at points $A$ and $S$. Let $J$ be the resulting point after reflecting $I$ w.r.t. line $BC$, and let $SJ$ meets the circumcircle of triangle $ABC$ at…
Vann
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