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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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Line passing through edges proof

Prove that there cannot be a line such that it passes through all the edges of a closed and odd-sided figure made of line segments. The line cannot contain any of the vertices of the figure as one of its points. I first set the condition that the…
Unaming Gaming
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How to find the angle which is in a trapezoid formed by two triangles?

The problem is as follows: The figure from below represents a quadrilateral $ABCE$. Using the information, $BE=BC$, $\angle\,BAC=\angle\,ADB=60^{\circ}$ and $DC=10\,m\,AE=10\,m$. Find the angle $x$. The alternatives given in my book are as…
Chris Steinbeck Bell
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what is length in euclidean geometry?

When we say an equilateral triangle with side length 3, what do we mean by length? Can we define length in euclidean-geometry?
user824904
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An explicit bijection between $R^n$ and $S^n\setminus \{$point$\}$

I am trying to formulate a bijective map $f_n:S^n\setminus (1,0,\dots,0)\to R^n$. I consider $S^n$ in $n$-spherical coordinates, that is, $S^n = \{(r,\phi_1,\dots,\phi_n)\in R^{n+1}\ |\ r = 1\}$, and $\phi_i\in [0,2\pi)$ for all $i$. It seems…
Jānis Lazovskis
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"Addable" similar n-gons

Let $X$, $Y$ and $Z$ be three similar n-gons with the following property: One can glue together $X$ and $Y$ along one of the edges to get $Z$. Examples would be rectangular triangles and also rectangles where the ratio of the sides is…
J Fabian Meier
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Find the value of BC

If $DE+EF=10$ and $(AC)(EF)=40$, find BC. I called $DE=x$, so $EF=10-x$, and we can find that $BD=\frac{20(5-x)}{x}$, but I couldn't do anymore in this question. I tried to find similar or equivalents triangles, but I didn't find. Can someone give…
Cavalo
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Area of a trapezium inscribed in a circle?

A circle, having center at $(2, 3)$ and radius $6$, crosses $y$-axis at the points $P$ and $Q$. The straight line with equation $x= 1$ intersects the radii $CP$ and $CQ$ at points $R$ and $S$ respectively. Find the area of the trapezium…
user791679
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Prove that there there exists a triangle similar to T.

Suppose all the points of euclidean plane are colored either red or blue. Given any triangle T, show that there exists a triangle similar to T in this plane for which all the vertices have the same color. I know a bit about Euclidean plane but…
Aditya Jain
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Optimization in Geometry: Q and R lie on $x$-axis and line $y=x$ respectively. Find Q if $PQ+QR+RP$ is minimum

Question: Let P be point $(5,3)$ on the coordinate plane. Let Q and R lie on $x$-axis and line $y=x$， respectively. Find Q if $$PQ+QR+RP$$ is minimum. I tried this question by taking any general point R on the line y=x and using triangular…
Shubhav Jain
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Prove that $\angle AQP = \angle ABC$

Given a triangle $ABC$ with $O$ as its circumcenter. Points $P$ and $C$ are the intersection points of the circumcircle of triangle $BOC$ and the circle with diameter $AC$. Point $Q$ lies on segment $PC$ such that $PB=PQ$. Prove that $\angle AQP =…
Vann
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What additional assumptions, if any, that we need to derive transitivity of parallel lines in plane geometry from strong version of Playfair's axiom?

My gf told me that his professor gave her homework. She says that the homework is not graded but will be asked in exam. It starts of with an axiom. Given a line and a point outside the line, there is one and only one line parallel to the original…
user4951
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Proposition 24 of Euclid's first book

I don't understand a statement in Euclid's proof of Prop. I24: Therefore the angle DFG is greater than the angle EGF.. Is it a consequence of what? This is the full proposition as stated here…
Michaelangelo Meucci
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Proof of proposition I.4 in Euclid's Elements ("Base angles of an isosceles triangle are congruent")

I have difficulties in understanding the 5th proposition of Euclid's elements in the first book: Proposition I.5. The base angles in an isosceles triangle are congruent. If the sides of an isosceles triangle are extended beyond the base, the angles…
Michaelangelo Meucci
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Coordinate geometry: finding the ratio in which a line segment is divided by a line

The question is: Determine the ratio in which the line $3x + 4y - 9 = 0$ divides the line segment joining the points $A(1,3)$ and $B(2,7)$. When I tried solving the question using section formula, which is: If $P$ divides the line segment…
Peeyush Kushwaha
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Extended mid-line, line through incircle touchpoints and angle bisector are concurrent - proof

In $\triangle ABC,$ the incircle touches $AB$ at $V$ and $BC$ at $T.$ $B'$ is the midpoint of $AC$ and $A'$ is the midpoint of $BC.$ $VT$ extended and $B'A'$ extended intersect at $P.$ Prove that $AP$ passes through the incentre, $I,$ of $\triangle…
Tom
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