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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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Very hard geometry problem.

Let $AB′C$ be a triangle with $\angle B′AC = 70^\circ$, $\angle ACB′ = 80^\circ$, $F$ is a point inside the triangle such that $\angle FB′A = \angle FCB` = 20^\circ$. Find $\angle AFB′$.
Pop Marius
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Euclidia isosceles triangle on tangent points 3.10

I've been struggling to come up with a solution for the Euclidia problem 3.10, creating an isosceles triangle from two given points on a circle. I can do it, but not in the # of steps given in the hints, for both the L and E solutions. Web search…
Laser Ray
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How to construct cube roots with an angle trisector

Is there a construction that, given lenght $l$, allows you to construct $\sqrt[3] l$? I think I have read that the trisector only allows for constructing roots of cubic equations with 3 real roots, whereof $x^3-l=0$ is not a example. What tools do…
FusRoDah
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What kind of isometry is a composition of a glide reflection with itself? Justify

Is there a simple algebraic proof? Thanks!
UH1
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prove the angle is 90 degrees

Given the trianle ABC,draw AD, where D is the middle of BC.If the angle BAD is 3 times the angle DAC and the angle BDA is 45 degrees,then prove that the angle BAC is 90 degrees. I tried to draw a parallel line to BA and compare congruent trianges…
user149368
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Integer distance between any two points in a plane

There is an infinite number of points in a plane. Devise an arrangement of these points in the plane in such a way that the distance between any two points in the plane is an integer. I realise that this is proved in the Erdős-Anning theorem;…
Saad Shaikh
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Curve through midpoints of the largest circles between two curves

I have two continuous differentiable curves in the interval $0 \leq x \leq 105$. I would like to fit the largest possible circle between the two curves for every x-value and make a curve through the center points. This figure shows an approximation…
Flinko
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Find $x$ in this $80^\circ$-$80^\circ$-$20^\circ$ triangle ($60^\circ$-$70^\circ$ variant)

Refer to the diagram and find x in degrees. My method is to let AB=1, and express AD and AE in terms of AB using sine formula. Then find DE using cosine formula. After that use cosine formula to find cos x. Finally x=20 degrees. However this method…
Ray Cheng
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Find the smallest segment contained between two non-parallel line passing a given point

The lines might as well be replaced by the sides of an angle. My gut tells me that it must be a segment perpendicular to the angle bisector but I can't justify it.
tighten
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Euclediean geometry proof

Let $ABCD$ be a square and $T$ a random point inside it. Let $A1,B1,C1,D1$ be points such that they belong to the lines $AT, BT, CT,$ and $DT$ respectively. Prove that $$\lvert A1B1\rvert\cdot\lvert C1D1\rvert=\lvert A1D1\rvert\cdot\lvert…
mathbbandstuff
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Construction Problem How to bisect an angle

Problem 3. To bisect a given angle (Figure 67), or in other words, to construct the bisector of a given angle or to draw its axis of symmetry. Between the sides of the angle, draw an arc DE of arbitrary radius centered at the vertex B. Then,…
Omicron
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Tangent Points for Common Tangent to Two Ellipses

This is somewhat similar to my other question here. Consider the two ellipses given by the equations \begin{equation} \frac{x^2}{2^2} + \frac{(y-1)^2}{1^2} = 1 \end{equation} and \begin{equation} \frac{x^2}{1^2} + \frac{(y-4)^2}{(1/2)^2} =…
Svend Tveskæg
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Significance of perpendicular line being shortest distance between a dot and line?

When I was 11 my maths teacher used the word perpendicular which I didn't understand so asked what it was. He then tried to make me figure out and he drew a diagram and I said is it the shortest distance possible. He seemed impressed and asked…
SRawes
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Euclidian embedding of lines

I'm looking for a way to convert a set of lines in R^3 into points in R^n so that distance between any pair of points points is a good approximation of the distance between corresponding pair of lines. Can anyone see a practical way of doing this?
Yaroslav Bulatov
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How are parallel lines defined in n-dimensional Euclidean space?

WolframMathWorld on parallel lines: Two lines in two-dimensional Euclidean space are said to be parallel if they do not intersect. In three-dimensional Euclidean space, parallel lines not only fail to intersect, but also maintain a constant…
philosopher
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