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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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Distance between a point and a line in space

I have two points $P_1(x_1, y_1, z_1)$ and $P_2(x_2, y_2, z_2)$ on a line, $L$, and another point $P_0(x_0, y_0, z_0)$. I want to find the distance between $P_0$ and $L$. Could someone help?
nkint
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Proof of Square Cube Law

Can someone offer a general proof of the square cube law of surface area and volume growth? For any fixed three dimensional shape it's clear how to proceed, but considering a general shape, how can one show the volume scales as the cube of the…
Bob
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Unlit region in a Room lined with Mirror

Mathematician Ernst Straus wondered if a room lined with mirror can always be lit with a single match. He (or someone else) discovered that in the following room, light shone from A can't reach B: The bad thing here is that light shone at middle…
finnlim
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Very naive questions in elementary geometry

I was wondering whether the following questions are difficult to solve : Consider a triangle ABC (defined in euclidean geometry). Let M be inside the triangle ABC such that the triangles AMB, AMC and BMC have same perimeter. What can be said about…
Curious
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Difficult geometry question involving Pythagoras theorem?

Hello mathematicians, I was given this question by my teacher and after spending a couple of hours looking over it have not been able to solve it. I understand it involves radians which I have attempted to learn through a few quick online courses,…
Monacraft
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Almost the Intercept Theorem

Consider the following figure, with $|\text{AD}|=|\text{CE}|$. If $|\text{AB}|=|\text{CB}|$, then $\text{AC}$ is parallell to $\text{DE}$ and $|\text{DE}|=\frac{|\text{AB}|}{|\text{AD}|}|\text{AC}|\le |AC|$, by the Intercept Theorem. If we don't…
Klaus
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Parallelogram and Areas

Consider the parallelogram $ABCD$. On sides $BC$ and $CD$ take points $E$ and $F $ respectively such that $\frac{BE}{EC} = \frac{CF}{ FD}$. If the segments $AE$ and $AF$ cut $BD$ at $K$ and $L$, show that $(AKL)=(BEK)+(DLF)$
user92596
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Trisecting a Triangle

Given a (non-degenerate) triangle $PQR$ in the Euclidean plane, does there exists a point $A$ in the interior of the triangle such that, the triangles $APQ$, $AQR$, and $ARP$ have same area? If it exists, is it unique? (I thought about this question…
Groups
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Two overlapping squares

$ABCD$ is a square. $BEFG$ is another square drawn with the common vertex $B$ such that $E,\ F$ fall inside the square $ABCD$. Then prove that $DF^2=2\cdot AE^2$.
lokesh sangabattula
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exercises for Euclid's Elements

Can you suggest some books with exercises related to Euclid's Elements, or to Euclidean Geometry, as an aid to an undergraduate course on Euclidean Geometry and its history? I need exercises that involve not complicated proofs.
peter
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Has anything further been done with Morley’s Miracle?

Or has it remained a terminal node at the frontier of mathematics?
Mike Jones
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What do the Purser's theorem says?

Mathworld's entry on Purser's Theorem says the following: Let $t, u$, and $v$ be the lengths of the tangents to a circle $C$ from the vertices of a triangle with sides of lengths $a, b$, and $c$. Then the condition that $C$ is tangent to the…
curious
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Pappus Chain Recursive Radii

Here is the equation I am asked to find. I have researched a lot into the Pappus chain, methods primarily involving circle inversion, but can only find examples of calculating the nth radius directly. Any hints/tips greatly…
Dennis
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The Euclidean norm ||r|| of a rotation

Please let me know what is the Euclidean norm of a rotation vector; and the difference from Euler angles. I met these terms in the following text ; although your answers do not need to respect the writer's intention at all , because you do not know…
seven_swodniw
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Must Basis of an Euclidean Space Be Ordered

Does the basis of an Euclidean space have to be ordered by definition? Or can be left unordered? I was also wondering about what is the morphism (i.e. the mapping that can preserve all the structures) on Euclidean spaces? Is it Euclidean…
Tim
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