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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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I've been trying to solve this particular geometry problem for 3 to 4 days but have finally given up

I know the solution that is 30 degrees here but I need to know the method so that I can extend it to a general solution. A method with linear equations will be very helpful. ABCD is a square. Isosceles triangle ABE has base angles measuring 15…
L Lawliet
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The formula for the point position rotated around an axis of a sphere

Please let me know the formula for the point position rotated around an axis of a sphere. In detail, I want to do as follows. Given: any point $p_1$ to decide the rotation axis ax of a sphere of ( radius r and center c); any point $p_2$ rotated…
seven_swodniw
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Does the proportion of the volume of the hypercube to the volume of the containing hypersphere tend to 1 as dimensions grow?

I read on this website that an "infinite dimensional" (limiting case of dimensionality) hypercube containing a hypersphere has a ratio of the hypersphere:hypercube volumes that tends to 0. Is it also true that a hypersphere containing a hypercube…
Zach O
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How to find the angle in a triangle when is given measure of a height and a side?

The problem is as follows: In a triangle $\triangle{ACB}$, $AC=20\,m$ it is also known $\angle{A}=2\angle{B}$ and the length of the foot of the height drawn from vertex $C$ to point $B$ is equal to $30\,m$ Find angle $C$. The choices given in my…
Chris Steinbeck Bell
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properties of symmetric difference of two set$|\Omega_1 \cap \Omega_2| |x_{\Omega_1} - x_{\Omega_2}| \le C(R) |\Omega_1 \Delta \Omega_2|$

how to show the following properties are holds in $\mathbb{R}^n$? for two bounded set $\Omega_1$, $\Omega_2 \subset \mathbb{R}^n$ and $\Omega_1$, $\Omega_2 \subset B_R$, $B_R$ is the Ball radius $R$ and centered at origion, then prop: $|\Omega_1…
blue denny
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Is there a collineation which does not preserve betweenness?

Consider the Euclidean plane $\mathbb{R}^2$. A collineation on $\mathbb{R}^2$ is a bijective function $f$ from $\mathbb{R}^2$ to $\mathbb{R}^2$ such that the image of every line under $f$ is also a line. Does there exist a collineation which does…
user107952
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Geometry proof using excenter & Radical axes

Q.Let ABC be a triangle with X, Y, and Z as excenters. Prove that triangle XYZ has orthocenter I and that triangle ABC is its orthic triangle. I am able to prove that ZBAI,XBIC and YCIA are cyclic by using excenter lemma. Then I can say that IB, IC…
Sauhardya Hazra
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Does there always exist a line that passes through two noncoplanar lines and a point

Given two noncoplanar lines $p$ and $q$, and a point $A$, does there always exist a line that passes through $p$, $q$ and $A$? This should be solved using Hilbert's axioms. Intuitively, that line doesn't always exist, but I don't know how to…
Ranko
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Are the lengths of the two sides of the given triangle equal?

For the above diagram, with A [2,1] and N [3,4], and angle AMN = 90 degrees, I would like to confirm if the length of AM is equal to MN. Also, what would be the length of MN. Here are the two threads where I have raised a similar query but getting…
Splendid Digital Solutions
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Proof of SAS congruency rule

I can find the proof(involving equations) of all congruence criterias of triangles except the S.A.S. rule . Everywhere, the proof of S.A.S. is given as they place one triangle on another triangle and find that all things coincides , which I can't…
Get_ Maths
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Prove that if $P, Q, R,$ and $S$ lie on a circle then the center of this circle lies on $XY$

Given circles $ω_1$ and $ω_2$ intersecting at points $X$ and $Y$, let $ℓ1$ be a line through the center of $ω_1$ intersecting $ω_2$ at points $P$ and $Q$ and let $ℓ_2$ be a line through the center of $ω_2$ intersecting $ω_1$ at points $R$ and $S$.…
PNT
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Proof based on geometry triangles

The angle between the median CM and the hypotenuse AB of right triangle ABC is equal to 30°. Find the area of ABC if the altitude CH is equal to 4. I am extremely sorry I tried but didn't make any desired progress that would help in solving the…
Death Champion
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Prove that the points $A, B, C, K$ are concyclic.

In scalene triangle $ABC$, let $K$ be the intersection of the angle bisector of $∠A$ and the perpendicular bisector of $BC$. Prove that the points $A, B, C, K$ are concyclic. $D$ is the midpoint of $[BC]$. Let $K’$ be the intersection of the angle…
PNT
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Prove that the reflections of the orthocenter (about the different sides) lie on the circumcircle.

Let $\triangle ABC$ be acute, $H$ is its orthocenter, and $(C)$ is its circumcircle. $E,F,D$ are the orthogonal projections of $H$ on $AC,AB,BC$. Let $H'$ be the reflection of $H$ about $BC$. Prove that $H'\in (C)$. I don't know if that's right but…
PNT
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Prove that $GD=GE$.

Let $ABC$ be a triangle, $F$ is a point inside the triangle such that $\angle ABF = \angle ACF$. $E$ and $D$ are the orthogonal projections of $F$ on $AB$ and $AC$, $G$ is the median of $BC$ , prove that $GD=GE$. There is a problem with my diagram,…
PNT
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