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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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How to find the point that is the minimum total distance from $N$ points in an arbitrary dimension?

Given a set of $k-\text{dimensional}$ points $S$, how do you find the point $p$ that minimizes the sum of the distances from $p$ to each point in $S?$
Gilad Felsen
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Does this proposition hold? About Euclidean Distance and Open Ball

About Euclidean space (n-dimensional) dn(x,y) is Euclidean distance between x and y. For any $a∈R^n$ and $r>0$ , $B(a,r):=\{x∈R^n:dn(a,x)0$ , $B(b,R):=\{x∈R^n:dn(b,x)
daㅤ
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Proposition about Euclidean distance and open ball

About Euclidean space ($n$-dimensional) $d_n(x,y)$ is Euclidean distance between $x$ and $y$. For any $a\in\mathbb{R}^n$ and $r>0$ , $B(a,r) := \left\lbrace x\in\mathbb{R}^n : d_n(a,x)0$ , $B(b,R)…
daㅤ
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Prove that the line passing through the intersection of the two diagonals of a parallelogram splits the side into two.

In the following diagram, giving we do not know that XZ is parallel to YV, prove that $WX=XY$. I have marked the values I do know are equal, but I am a bit stuck about continuing.
Jamminermit
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Why are the shortest distances in Euclidean geometry not taxicab?

I am of course well aware that distances in Euclidean geometry are calculated from the Pythagorean theorem. This is more of a conceptual question. My question may also be formulated as follows: If we approach the diagonal of a square (of side length…
Thomas Jungers
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Complex Euclidean Geometry Question

Let w be the incircle of a fixed equilateral triangle ABC. Let l be a variable line that is tangent to w and meets the interior of segments AC and BC at P and Q, respectively. A point R is chosen such that PR = PA and QR = QB. Find all locations of…
user744468
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Proving the converse of Ceva's theorem

This is what the converse of Ceva's theorem states. Suppose $ABC$ is a triangle, and let $AD$, $BE$, $CF$ be the three cevians such that $$\dfrac{BD}{DC}\cdot\dfrac{CE}{EA}\cdot\dfrac{AF}{FC}=1$$ Then $AD$, $BE$, $CF$ are concurrent. The proof…
Aritriya Mukhapadhayay
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Euclid's proof of Side-Angle-Side equality

I was looking at Euclid's proof of the triangle Side-Angle-Side equality, and I'm not really convinced by it. The first step of the proof is to place the first triangle onto the second one, but why can we do this? I agree that we are able to…
David
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Verifying a line is tangent to a circle circumscribing a triangle

$\triangle\mathit{ABC}$ is an isosceles triangle with legs $\overline{\mathit{AB}}$ and $\overline{\mathit{BC}}$. $\omega$ is the circle circumscribing the triangle, and $D$ is the intersection of the tangent lines to the circle at $A$ and at $B$.…
user74973
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Prove that the diagonals of a trapezoid, one of whose bases is twice as long as the other, divide its midsegment into three equal parts.

I am not sure where to start. A hint would be great to start off this problem.
Mikal Jardin
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Center Selection Problem: Covering radius

I am trying to understand the following text which defines a greedy algorithm for center selection problem: It would put the first center at the best possible location for a single center, then keep adding centers so as to reduce the covering …
zak100
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Stewart's theorem inefficient proof

Question - In triangle ABC, point D divides BC in n/m(=BD/DC) ratio. AE is the altitude from A to BC. Prove that (AB^2)*m + (AC^2)n = m(BD^2) + n*(CD^2) + (m+n)*(AD^2) I was trying to prove Stewart's theorem by simply expanding LHS(AB^2*m + AC^2*n)…
skallu
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A geometric result involving incentre and excentre

Let ABC is a triangle and $I_c$ and $I_b$ be excentres opposite to vertex C and vertex B respectively. Join A,$I_c$ and $I_b$. The line touches circle again at point Q .We have to prove that Q is the midpoint of line segment $I_bI_c$.I tried to…
Mathematical Curiosity
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Betweenness proof relating to false proof for "all triangles are isosceles" / danger in diagrams

There are at least two questions on this site related to the (somewhat famous) false proof that all triangles are isosceles as presented for example in M. J. Greenberg's 'Euclidean and Non-Euclidean Geometry'. The fallacy itself can be read on this…
user77970
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Euclidean construction of one degree angle

We can construct 15 degrees by bisecting 60 degrees twice. We can construct 37 degrees by constructing a right angled triangle with sides 3 and 4 since tan 37 is 3÷4 We can also similarly construct 53 degrees When we have constructed 37 and 53 we…
Siddhartha Singh
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