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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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3 reflections leads to glide

In a previous problem I asked about the notation used here.. I'm still not sure how to show it even though I now get what it's asking. The following is an exercise from The Four Pillars of Geometry. 3.6.5. Show that the reflections in lines $L$,…
ttt
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Prove that $l_1$ and $l_2$ are parallel if and only if $a_1b_2-a_2b_1=0$

For $b_1$ and $b_2$ non-zero, consider the lines $l_1=\{(x,y) \in \mathbb{R}^2 | a_1x + b_1y + c_1=0\}$ and $l_2=\{(x,y) \in \mathbb{R}^2 | a_2x + b_2y + c_1=0\}$. Assuming I only know Euclid's postulates and the definition that lines are parallel…
flamingohats
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Geometry question

In the given figure , AP, BQ and CR are perpendicular to line AC. And AP=$x$ , BQ=$y$, CR=$z$ then find the value of $\frac{1}{x} + \frac{1}{z}$ in terms of $y$. I have no idea how to solve it. But the limitation is this is to do with the use of…
curious_mind
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Euclidean Geometry with Feet of Perpendicular

Let ABC be a triangle with angle B = 90 degrees. Let D, E be points on AB such that BD = BC and AE = AC (they lie in the order B-E-D-A). Let F, G be the feet of perpendicular from E and D onto BC and AB respectively. If EF = 12 And DG = 11, what is…
Wafi Marzouq Mohammad
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Angle between the orthocenter and incenter

From I am asked to get the value of $\theta$ in terms of $\alpha$ and $\beta$. Quick geogebra "shows" it is the difference, but how do you show it mathematically? All the properties I have about it are that the incenter is the intersection of the…
chubakueno
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Triangle inequality question.

For triangle ABC, there is a point X such that B-A-X. There is any point P on bisector of exterior angle CAX. Prove that PB+PC>AB+AC. I have tried for many hours, but i have no idea how to solve it. actually, i have shown PB>AB. This is a question…
curious_mind
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Bisecting an angle with a right-angled bent ruler (that can coordinate the vertex and both edges)

This thesis claims that you can construct any point constructible with ruler and compass with only a "bent ruler" -- that is, a pair of moveable rays joined at a specific angle. "Euclidean constructions: alternate tools to the traditional compass…
DavidButlerUofA
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Equilateral pentagon with three adjacent equal angles is regular.

Showing that equilateral pentagon with three adjacent equal angles is regular using only euclidean geometry. Hypothesis: sides $AB=BC=CD=DE=EF$ and angles $\angle EAB=\angle ABC=\angle BCD.$ Thesis: angles $\angle ABC=\angle BCD=\angle CDE=\angle…
Jess
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Geometry problem about square

We need to solve this problem without using trigonometry. KLMN square is inside ABCD square. Prove that midpoints of the segments AK, BL, CM, and DN are vertices of a square. KLMN can be situated anywhere inside ABCD. I think that we are going to…
Zhenya Karapetyan
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How to prove this problem by using similarity of two triangles?

I have this problem: Let be given a right triangle $ABC$ $(AB
minhthien_2016
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How to find a point or a plane which is equidistant from n planes in an n-dimension space (emphasis on point equidistant from "n planes")

I am working on an optimization problem where I have converted the physical system into a set of bounding hyperplanes. To keep it simple, let's say I have three planes in $R^3$. I am trying to find the equation of a point or a plane that is…
Nisharg
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Importance of I.45 in Greek mathematics

I’ve been going through Euclid Book I and I’ve got to proposition 45. This is a construction that allows the area of a rectilineal figure to be converted into a parallelogram with given angle and side. Reading Heath’s Thirteen books of Euclid’s…
rhody
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Are the theorems of Euclidean geometry preserved under dilations?

I know that the theorems of Euclidean geometry are preserved under rotations, reflections, translations, and compositions of the former three. But what about dilations? Intuitively, the theorems should not change if we measure distances using inches…
user107952
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Circle lettering in Book 1, Proposition 1 of Euclid's elements

I've been looking at Euclid's Elements, and I have a question about Book 1, Proposition 1. This is where Euclid constructs an equilateral triangle. The proof is straightforward, but I have a question about how Euclid identified the circles he uses…
rhody
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bisectors of exterior angles in triangle

In triangle ABC, the bisectors of the exterior angles B and C meet at H. Show that AH is the bisector of the angle BAC. I was trying to look at some exterior angles in some triangles but I can't find a way to link those two angles that supposed to…
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