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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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Simple geometry question, to be proved without trigonometry

In triangle $\triangle ABC$, ray $AD$ is a bisector of angle $A$, which intersects $BC$ at $D$. Also given are that $AC$ = 4 cm, $AB$ = 3 cm and $\angle A = 60^\circ$. Find the length of $AD$. This is a simple geometry question. I had a trivial…
curious_mind
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Why is proving converses of propositions (specifically in Euclid) important?

Why does Euclid demonstrate several props by their converse or through a reduction to the absurd? In what way is proving that the converse of the prop is absurd preferable to proving constructively what the prop sets out to do.
Tom
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Euclid's Elements Book I, Proposition 17 and Euclid's 5th postulate

I was going through Euclid's elements when I noticed Book I, Proposition 17, which states that: In any triangle the sum of any two angles is less than two right angles. And also Euclid's 5th postulate which states that: If a straight line falling…
Areen Rath
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Why does proposition 1 in Euclid's Elements not end with "QEF"?

I am reading from Heath's translation of Euclid's Elements. Most propositions either end with "QED" or with "QEF", when the first one is used for proving propositions that are not constructions and the second one is for constructions. However, in…
Hilbert
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is line an Euclidean space?

On one hand, in order to be a Euclidean a space should be equipped with a concept of "an angle". And the angle on a line has only two values (0 and 180 degrees), while in higher dimensional spaces it can have a continuum of values. So, is it…
sixtytrees
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If two arcs are equal, then angle subtended by both arcs on a fixed point are equal. How?

[AHSME 1971] Quadrilateral $ABCD$ is inscribed in a circle with diameter $AD = 4$. If sides $AB$ and $BC$ each have length $1$, then find $CD$. I tried looking through the solution of this, and couldn't understand the following part: Since $AB =…
acelixis
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Are sets with at most one point collinear?

In Euclidean geometry, every set with $2$ points is collinear. My question is, are singleton sets and the empty set also collinear? I think it depends on the exact definition of collinear.
user107952
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Distance between taking the shortest paths vs taking the longest paths

Assume we have a finite set of points A in $\mathbb{R^2}$ or $\mathbb{R^3}$. Is it true that if we start from a fixed point and travel to the closest unvisited point, the sum of these distances will be no greater than the sum of lengths if we always…
user90237
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Computing the distance from a point to an intersection of two hyperplanes

Consider the hyperplanes $\{x | P_1^T \cdot x + q_1 = 0\}$ and $\{x | P_2^T\cdot x + q_2 = 0\}$ in $\mathbb{R}^n$. Let $C \in \mathbb{R}^n$ and we want to compute te distance from $C$ to the intersection of these hyperplanes. Solution Let $x = C + a…
C Marius
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Let $ABCD$ be a convex quadrilateral with $AD = BC$ and $∠A + ∠B = 120°$ . Prove triangle formed by midpoints of $CD$, $AC$ and $BD$ is equilateral.

Let $ABCD$ be a convex quadrilateral with $AD = BC$ and $∠A + ∠B = 120°$. Let $E$ be the midpoint of the side $CD$ and let $F$ and $G$ be the midpoints of the diagonals $AC$ and $BD$, respectively. Prove that $EFG$ is an equilateral triangle. $E$…
Frosty
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Proof that two segments don't intersect

I'm struggling with the following elementary problem from Euclidean Geometry which is the last piece within a certain framework of a proof that congruent alternate interior angles implies parallel lines. I am trying to do this all without any notion…
user2951662
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question about possibility of 3 given functions of lengths of a triangle, themselves forming a triangle.

x = f(a,b,c), y = g(a,b,c), and z = h(a,b,c) if a,b,c are sides of a triangle, then x,y,z are also sides of a triangle. How can you write sets of functions f,g and h?
Seetha Rama Raju Sanapala
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An easier solution to Euclid's Book I, Proposition 5?

Proposition 5 Does not Common Notion 3 (If equals are subtracted from equals the remainders are equal) solve Proposition 5 all on its own? Lines AB and AC are equal. The base angles ABC and ACB are equal. Extending lines AB and AC, and plotting…
Squinter
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How can one translate an orthogonal vector along parallel lines to a desired location?

Question: How can one translate a perpendicular along parallel lines to a given point? Here is the problem. Given a curve $\mathscr{C}$ (blue), a point $A\in \mathscr{C}$, the tangent line $T_A$ (green) at $A$, the line $L_A$ (orange) going through…
Nap D. Lover
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Equivalence of Clavio's and Clairaut's axioms

We have the following axioms: Clairaut's Axiom. If $AB$, $AC$ and $BD$ are segments such that $AC$ and $BD$ are congruent and both perpendicular to $AB$, then the angles on $C$ and $D$ are both right. In other words, $\square ABDC$ is a…
Superdivinidad
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