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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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Finding distance from point to line

Knowing the position of 3 points($A, B, C$) , how can I get the distance from $A$ to the line $\overline {BC}$ if I know the angle?
user80458
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Colinearity of five points and parallelism of five lines in a geometric figure involving two regular pentagons and three squares

This is a follow up to my solution to the question of peta arantes. In that solution, i am showing a proposition, which solves the cited question in a more general situation, starting with points $A,B,D$ in general position. However, during the…
dan_fulea
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Deduction of natural numbers from Hilbert axioms

I state that I am not an expert in formal systems, but I did something as a "self-taught" because I am passionate about it. My question is the following: since geometry hilbert axioms are equivalent to the real numbers axioms from the primes one…
user273366
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Collinearity of centres of Apollonius circles in a scalene triangle

In scalene triangle $ABC$, $AD$ and $AD'$ are internal and external bisectors of $\angle A$, respectively, meeting $BC$ at $D$ and $D'$, respectively. If $A'$ is the midpoint of $DD'$, and similarly $B'$ and $C'$ are the corresponding on $CA$ and…
Karthik Nambiar
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Curious basic Euclidean geometry relation

I was plotting some equations and I got with the curious relation If we build the triange Such that it follows the following relation: $$AD=a$$ $$DB=b$$ $$AC=ak$$ $$CB=bk$$ Then when we vary $k$ from the smallest possible value to the greatest…
chubakueno
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Prove or disprove there does not exist any triangle $\triangle {ABC}$ in which the line $IG$ is perpendicular to the line $IC$

The problem, asked 14 Feb at 18.18 by user @r ne, (see figure $1$ below) would appear to have been abandoned because of some adverse comment about “How to ask a good question”. The attached figure $2$ below shows the triangle with sides $14, 15$ and…
Piquito
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Triangle with integer sides, and centroid-incenter $\perp$ incenter-C, what is the perimeter of triangle?

See the figure: All the sides of $\triangle{ABC}$ are positive integers. The greatest common factor of $a$ and $b$ is $2$. $G$ and $I$ are the centroid and incenter of the triangle. $GI \perp CI$. What is the perimeter of the triangle?
r ne
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Thirteen properties of Euclidean Geometry

I'm reading "Taxicab Geometry: Adventure in Non-Euclidean Geometry" by Eugene F Krause where he mentioned: A geometry is called Euclidean if it satisfies the following thirteen properties: Given any two points there is exactly one line containing…
athos
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Inversion problem for four points

If $A$, $B$, $C$, $D$ are four non-coplanar points, then show, by inversion with respect to any one of those points, that the sum of any two of the products $BC\cdot AD$, $CA\cdot BD$, $AB\cdot CD$ is greater than the third. I have shown that if…
D. Spencer
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Finding a cosine in a cube

Given a cube $ABCDA'B'C'D'$ and S,T,R,N the middle of $AB, DD', DC,$ respectivley $BC'$. Find the cosine of the angle of $TS$ and $RN$. I tried to use paralelism and perpendicularity theorems, then the cosinus theorem, but I didn't match very well…
mmm
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How can the product of $\sqrt{x}$ be negative?

I've recently implementing my own raytracer in C. I was reading up some paper on ray-sphere intersection from http://www.cs.unc.edu/~rademach/xroads-RT/RTarticle.html and this came up: $d = \sqrt{r^2 - (c^2 - v^2)}$ To determine whether an…
Spade 000
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How to find the angle formed by two joined triangles when they are multiples?

The problem is as follows: The figure from below shows two triangles. Assume $PR=QS$. Using this information find $x$. The alternatives given in my book are as…
Chris Steinbeck Bell
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How to find the angle formed by an isosceles triangle next to another one?

The problem is as follows: Using the figure from below: Find the unknown angle indicated as $x$. Assume $AD=BC$ and $BD=DC$ The alternatives given in my book are as…
Chris Steinbeck Bell
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When is the hexagon formed by the feet of perpendiculars and the midpoints of sides regular?

In a $\Delta ABC$, consider the hexagon $m_{a}f_{a}m_{b}f_{b}m_{c}f_{c}$ where $m$ and $f$ stand for the midpoint and foot of perpendicular of the respective sides. Now it can be shown quite easily, that $$m_{a}f_{a}= |\frac{b^2-c^2}{2a}|$$ With a…
kodyv
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How to find the distance covered by a light in a flat crystal?

The problem is as follows: The figure from below shows a lactose crystal which is about to be studied for its refraction index, for this purpose an orange light beam is focused by means of a lens over a LED. The crystal was cut as a thin plate. The…
Chris Steinbeck Bell
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