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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.

9328 questions
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Does this triangle-area theorem have a name?

Given two distinct parallel lines and two distinct fixed points on one of the lines and a point that can vary on the other line. Then the areas of all the triangles formed by those 3 points are all equal. – Does this theorem have a name? I am not…
user584285
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Prove that $PX+PY=PZ$

In triangle $ABC$, let $D$ and $E$ be the feet of the angle-bisectors of angles $A$ and $B$. Let $P$ be a point on $DE$. Let the feet of the perpendiculars onto $CB$, $AB$ and $CA$ be $X$, $Y$, $Z$ respectively. Prove that $PX+PY=PZ$. So far…
Plato
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Collinear intersection points of diagonals in a regular heptagon

In a regular heptagon, all diagonals are drawn. Then points $A$, $B$, and $C$ in the diagram below are collinear: (If the vertices of the pentagon are $P_1, \dots, P_7$ in clockwise order, then $A = P_1$, $B$ is the intersection of $P_2 P_5$ and…
Misha Lavrov
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Points closest to the center of the cube

A set of $8$ points in an unit cube is given, such that the distance between any two points belonging to the set is not less than $1$. Find the distance between the center of the cube and the closest to it point from the set. Intuitively, I think,…
prometheus21
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Euclidea 3 9.8 Chord Trisection

Construct a chord of the larger circle through the given point (on circumference of larger circle) that is divided into three equal segments by the smaller circle (circles are concentric) I'm having trouble finding a method to solve this geometric…
Jeremy Corrin
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Dividing a disk into $7$ equal pieces with $3$ line segments

Can you divide a disk into $7$ pieces of equal area, with $3$ line segments? (You can surely divide it into $7$ pieces, but could those have equal areas?) (This question was left unanswered at another forum. I can see with some visual arguments…
AgCl
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Size of a point.

I know this may sound too simple or maybe too absurd to discuss, but I am having a hard time visualizing a point in space! In Euclid's Elements a 'Point' is defined as Something which has no part. Now, any geometrical figure viz. Line…
Señor Pérez
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Polar of a non centred ball.

Recall that the polar of a set $A\subset\mathbb{R}^n$ is the following set: $$A^{\circ} = \lbrace c \in \mathbb{R}^{n} |\; \langle c,x \rangle \leq 1 \quad \forall x \in A \rbrace$$ where $\langle \cdot, \cdot \rangle$ is the usal scalar…
Gilles Bonnet
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Why are these three lines concurrent?

Consider a triangle $ABC$ with incentre $I$ and let $AI \cap BC=D$. Let the incentres of $\triangle ACD$ and $\triangle ABD$ be $E$ and $F$ respectively. Prove that $AD$, $BE$ and $CF$ are concurrent. This is part of a larger problem I am trying to…
John Marty
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Square inside triangle

Is there a triangle $ABC$ having following properties? : $ABC$ is not right angled $d(A,B)\ne d(A,C)$ If one draws the largest possible square such that one of the side of square is a subset of $AB$ and rest two vertexes are on sides $AC$ and $BC$…
guest
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rotate vector around another vector

If I have vectors $a = (1,0,0)$, $b = (0,1,0)$, and $c=(0,0,1)$ and I want to rotate them counterclockwise at rate $r$ rad/sec around vector (1,1,1). What are the formulas for $a, b, c$?
SuperGuy
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How many vectors are needed to define a plane in n dimensions?

How many vectors are needed to define a plane/hyperplane in n-dimensional space? In 3 dimensions, if there are 2 vectors with tails at the origin and the heads in differing locations (and the vectors aren't parallel), that information is sufficient…
Lee
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Congruency and congruent triangles

Given that $\Delta\,ABC$ is an isosceles right triangle with $AC = BC$ and $\angle{ACB}= 90^\circ$. D is a point on AC and E is on the extension of BD such that $AE \perp BE$. If $AE = \frac{1}{2}BD$, prove that BD bisects $\angle{ABC}$. Since $AE…
Mathematical Physicist
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Parallel postulate from Playfair's axiom

Parallel postulate: 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…
user5402
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Regular Pentagon, Geometry

Can someone help me to find the value of $x$? I wish I could share my attempted solution but I really couldn't develop something interesting to share. I know that the value of the internal angle are $108^°$, because the sum of the internal angles of…
Gustavo Gabriel
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