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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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Find Angle Between Planes in Space from angles of line through them.

Find the angle between two planes in space. The only givens are the angles that each plane makes with a line through them both.
JLA
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Calculate the area of the triangle

Here is a cute little problem. In the diagram below $ABC$ is a right triangle, $\angle ABC$ is a right angle. The blue region inside the triangle is a rectangle. Given that point $G$ is both a corner of the rectangle and the centroid of the…
blackened
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What Type of Curve Is This?

What curve is the longest distance between points a and b such that (i) one gets closer to point b at all times; (ii) it can be graphed as a function; and (iii) it is symmetrical? I would think each side approaches the interior perimeter of a circle…
Aldante
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Simple problem in Euclidean Geometry -- Find the radius of a circle

A student of mine brought the following question to my attention. I am currently not able to solve it, any help would be appreciated. It should be a simple circle theorem that I have now forgotten. Question: Let $AB$ be a chord of a circle with…
AmorFati
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Prove that triangle EDM is an isosceles right angel triangle

In the diagram below triangle ABE is an isosceles right angle triangle and triangle ADC is an isosceles right angel triangle and M is in the middle of BC prove that triangle EMD is an isosceles right angel triangle
user659249
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Prove that $HI \perp BD$.

$I$ is the incenter of $\triangle ABC$. The perpendicular bisector of $CI$ cuts $AI$, $BI$ and $CA$ respectively at $D$, $E$ and $F$. The line that passes through the midpoint of $IF$ and is perpendicular to $BC$ intersects the line that passes…
Lê Thành Đạt
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Perimeters of isosceles triangles

Two non-congruent integer sided isosceles triangles have the same area and perimeter. The ratio of the lengths of the bases of the two triangles is 8:7. Find the minimum possible value of their common perimeter. I have tried substituting the area…
Hector Lombard
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Mean distance vs. distance to mean (L2 norm)

Intuitively, I feel like the following should hold, but I fail to prove it: $\lVert x^*- \frac 1k \Sigma_{i=1}^k w_ix_i\rVert_2 \overset ?= \frac 1k \Sigma_{i=1}^k w_i \lVert x^*- x_i\rVert_2$ Where $x_1,...,x_k, x^*\in\mathbb R^d$ I.e. is the…
Shay BE
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Existence of a triangle with sides partitioned in particular ratios by inscribed circle

Is there a triangle with sides that are partitioned into line segments of ratios $3:2$, $3:5$, and $10:9$ by the points of tangency of its inscribed circle? By Ceva's Theorem, if a triangle has sides of lengths 20, 16, and 19, the cevian between the…
A gal named Desire
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A line connecting an interior and an exterior point of a circle should intersect the circle at some point

How can someone prove in Euclidean geometry that the statement "A line connecting an interior and an exterior point of a circle should intersect the circle at some point" follows from the axioms of Hilbert or Birkhoff? I cannot find any relevant…
Sumac
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Plane Geometry related to Circle

The internal bisector of angle A of triangle ABC meets the circumcircle in D. If DE and DF are the perpendiculars to AB and AC respectively from D. Prove that AE is arithematic mean of AB and AC .
Rishabh Shetty
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stereographic projection $\sigma:\mathbb{R}\rightarrow S^1$

The line which connects a Point $x\in\mathbb{R}\subset\mathbb{C}$ with $i$ intersects the unit circle in one Point. This Point shall be called $\sigma(x)$. Now I Need to find a closed form for $\sigma(x)$. There are hints and a solution but I don't…
RM777
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Points, planes etc. embedded in higher-dimensional Euclidean spaces

A common set of definitions for a plane is: three non-collinear points a line and a point not on that line two distinct but intersecting lines two parallel lines. Is it possible to provide a generic set of definitions for higher-dimensional…
Sky Wilshaw
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Proof that the middle line of a trapezoid bisects any segment that joins points from its bases

I know that the middle line of a trapezoid is the line that joins the midpoints of the non-parallel sides, and that it is parallel to the bases of the trapezoid. How do I prove that the middle line of a trapezoid bisects any segment that joins…
TonyStank
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Algorithm for Euclidean K-Center Problem

I am studying the Euclidean $k$-center problem. This paper proved that the problem is NP-hard for any arbitrary $k$. However, in this paper authors provided an algorithm for finding solutions for the $k$-center problem in $O(n^{O(\sqrt{k})})$ time.…
Mateo
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