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.

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Are there two triangles with equal angles and two equal sides which are not congruent?

Are there two triangles with equal angles and two equal sides which are not congruent? I think it is impossible.
chen h.
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What is the range of longitude/latitude coordinates given a radius of area on Earth that I want to cover?

Given a pair of coordinates, I would like to have a range for both long/lat, that would cover, say 100 meters. How can I do this?
nubela
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euclid's 13th proposition

Euclid's 13th Proposition goes as : Proposition 13. If a straight-line stood on a(nother) straight-line makes angles, it will certainly either make two rightangles, or (angles whose sum is) equal to two rightangles. I am annoyed because I find…
saibal
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Is it possible for Euclid's theorem to generate 3 4 5?

I am writing an algorithm to generate pythagorean triplets and I was going to use Euclid's theorem, however I have been unable to make it generate the first pythagorean triplet namely (3,4,5). Is this a limitation of the theorem? Does this occur for…
stmfunk
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Proving that Reflections of Orthocenter over Midpoint and Opposite Side Lie on Circumcircle

Let $H$ be the orthocentre of an acute $\Delta ABC$, as in figure. Let $X$ be the reflection of $H$ over $\overline{BC}$ and $Y$ be the reflection of $H$ over the midpoint of $\overline{BC}$. Show that $X$ and $Y$ lie on the circumcircle of $\Delta…
Harshul
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If X, Y, Z are feet of the perpendiculars from the centroid G of $ABC$ upon the sides of AB,BC, CA. Prove $[XYZ] = 4\Delta^2(a^2+b^2+c^2)/9a^2b^2c^2$

If $X$, $Y$, $Z$ are feet of the perpendiculars from the centroid $G$ of $\Delta{ABC}$ upon the sides $BC$, $CA$, $AB$, prove that $[XYZ] = \frac{4\Delta^2(a^2+b^2+c^2)}{9a^2b^2c^2}$ where $\Delta$ is the area of $\Delta{ABC}$ I noticed XCGY is…
Frosty
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Is there any distance function such that $g(\mathbf{b}+\mathbf{c}) = g(\mathbf{b}) + g(\mathbf{c})$?

Suppose that we have $$\mathbf{a}_i - \mathbf{a}_j = \mathbf{b} + \mathbf{c},$$ where all variables are of the same dimension (i.e., $n$ dimensional). If we let $g(\mathbf{a}_i-\mathbf{a}_j)$ denote the squared Euclidean distance between…
Ron Snow
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Geometry Problem on Finding Angles

In an isosceles triangle ABC,AB=AC,P and Q are points on AC and AB respectively such that CB=BP=PQ=QA.Then prove that angle AQP=900/7
sangabattulalokesh
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What is an arbitrary point?

Firstly, how is an arbitrary point defined in plane geometry? I came across many proofs which use an arbitrary point to prove something which is true for all points. For eg: Prove that the tangent to a circle is perpendicular to the line passing…
PandaScientist
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A formula for a dilation of the real plane with a non-origin center

I know that the formula for a dilation in the real plane $\mathbb{R}^2$ with center the origin $(0,0)$ is $(cx,cy)$, with $c \neq 0$. What is the formula for a dilation in the real plane with center an arbitrary point $(a,b)$?
user107952
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Is there a subset of the real plane such that every line intersects that subset exactly once?

Consider the real plane $\mathbb{R}^2$. Does there exist a subset $S$ of the real plane, such that every line $l$ in $\mathbb{R}^2$ intersects $S$ at exactly one point?
user107952
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Construct reflection of a point on a line with respect to a parallel line knowing a point and its reflection

Below are two lines $(D)$ and $(\Delta)$. We are given a point $M$ on $(D)$ and its reflection wtr $(\Delta)$. If $A$ is another point on $(D)$, it is asked to construct the reflection of it wtr $(\Delta)$ Using straightedge only. I tried to use $J…
Karim Kamil
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How to find the length of a segment which connects two triangles in a rhomboid?

The problem is as follows: The alternatives in my book are as follows: $\begin{array}{ll} 1.&\textrm{1 cm}\\ 2.&\textrm{3 cm}\\ 3.&\textrm{5 cm}\\ 4.&\textrm{6 cm}\\ \end{array}$ I was only able to spot on: $\triangle APT$ then $AT=…
Chris Steinbeck Bell
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How to get the angle formed by an hexagon which has two lines on it?

The problem is as follows: In a regular hexagon $ABCDEF$ a point $G$ is located on $CD$ such that $AG$ and $BE$ intersect at point $R$. If $\angle AGD = 110^\circ$. Find $\angle BCR$. The alternatives given in my book…
Chris Steinbeck Bell
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Is shrinking a square to a point reversible?

From Wikipedia, "Geometric points do not have any length, area, volume or any other dimensional attribute. A common interpretation is that the concept of a point is meant to capture the notion of a unique location in Euclidean space." Suppose it…
vengy
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