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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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Given the figure. Prove that $CA=2 \cdot EH$

In the figure $\overleftrightarrow{BE} \parallel \overleftrightarrow{AC}$, $\overleftrightarrow{FG} \parallel \overleftrightarrow{AB}$ and $\frac{AB}{DB}=\frac{BC}{BE}=\frac{CB}{CG}=\frac{CA}{CF}=4$ Prove that $CA=2 \cdot EH$ I know that I need to…
HighSchool15
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Geometry and van Aubel's theorem

Given an arbitrary planar quadrilateral, place a square outwardly on each side, and connect the centers of opposite squares. Then van Aubel's theorem states that the two lines are of equal length and cross at a right angle. How can I prove that the…
user504516
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Distance between two functions to create a third function,

In this question I am using the euclidean metric to determine the distance between two points. I want to make a function $f(x)=$ the minimum distance between $y=x$ and $y=e^x$ at each given point x, is there an efficient way of doing this? Second…
shai horowitz
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Chord with the smallest length

If I have a point K inside a euclidean disc what will be chord (chord which goes through the point K) with the smallest length. I think it will be chord which is perpendicular to the diameter, which goes through the point K, but I don't know is this…
Юрій Ярош
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the trajectory of points whose powers of point to two circles have constant ratio

Let $\omega_1$ and $\omega_2$ be circles with respective centers $O_1$ and $O_2$ and respective radii $r_1$ and $r_2$, and let $k$ be a real number not equal to 1. Prove that the set of points $P$ such that $$ PO_1^2 - r_1^2 = k(PO_2^2 -…
zhaofeng-shu33
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Why is it true that the angle $\angle AIC = 90^{\circ}+ \frac{\angle ABC}{2}$

Consider the following picture (borrowed from the web). It is a well-known fact that I most recently saw on page 12 of Coexeter's *Introduction to Geometry that the angle $\angle AIC = \frac{\angle ABC}{2} + 90^{\circ}$. I am having trouble seeing…
user135520
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Max Euclidean Distance between two points in a set

Given a set of Euclidean Vectors with $N$ dimensions, whose distance from a Euclidean Vector, $R$, is less than some Constant, $C$. Can the max distance between any two vectors in the set be determined? I have been searching for some sort of proof…
Steven Feldman
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Can a straight line be considered a part of a circle?

Say we have a circle of infinite radius. If we zoom in infinitely on its perimeter, we should end up looking at a straight line. Intuitively. But, such a line, I believe, should have a curvature of $1/∞$. But a straight line is defined to have $0$…
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a line through $2$ points

I remembered a question I once heard and never found the answer, maybe you can help me with that: given the set $S$. $S$ is a set of points on $\Bbb R^n, n\ge2$, which is finite and has more than $2$ points in it. prove or disprove the following…
ℋolo
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Pythagoras and its converse

I am currently studying euclidean geometry and there's is a problem that really boggles me. Actually, even knowing the answer I cannot figure how to "draw" the problem. Here it is: $ABC$ is a triangle right-angled at $A$: and the sides $AB, A$C are…
le sram du bled
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Find $S_{PCD} $ given $S_{ABCD},S_{ABP},S_{BPC}$

In the following figure,$ABCD$ is a paralleogram in which point $P$ lies in triangle $ABD$ and the areas of $ABCD,ABP,BPC$ are known to be $s_1,s_2,s_3$ respectively. Find $S_{PCD}$ I think the number of unknowns in this problem is more than…
Hamid Reza Ebrahimi
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In square $ABCD$ find angle $\angle DFA$ given that...

In square $ABCD$ point $E$ lies on $BC$ and point $F$ lies on $DE$ such that: $DF=BF , EF=EB$.Calculate the measure of angle $DFA$. It's easily seen that $\angle BFE=2\angle BDF$.But I can't go on...
Hamid Reza Ebrahimi
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Find a point, using the differences between distances from 3 other points

I have 3 points on a 2D plane ($A$, $B$ and $C$), with known co-ordinates $(x_A, y_A)$, $(x_B, y_B)$ and $(x_C, y_C)$. I need to find the co-ordinates $(x_Z, y_Z)$ of point $Z$. I know the differences between each pair of distances from points $A$,…
Topper1
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Relation involving lengths determined by a line (through the incenter?) of a triangle

From "A Beautiful Journey through Olympiad Geometry": Part II. Problem 2. Let $I$ be the incenter of $\triangle ABC$. Let $\ell$ be a line [through $I$?] parallel to $\overline{AB}$, that intersects the sides $\overline{CA}$ and $\overline{CB}$ at…
user459947
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Does this $n$-Ball 'grow' out of the unit box?

Let $K_n(r) = [-r,r]\times[-r,r]\times\cdots\times[-r,r] \in\mathbb{R}^n$ be the $n$-Box of edge length $2r$. I think we can uniquely partition $K_n(1)$ into $2^n$ disjoint n-dimensional cubes, each of which are translates of $K_n(1/2)$. If so, then…
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