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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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Maximal area of triangle if angle and opposite side length is known

Two lines $l_1$ and $l_2$ intersects at point $A$ such that the angle they intersect is $\alpha$. A line segment has endpoints $B$ and $C$ in the lines $l_1$ and $l_2$, respectively, and $|BC|=l$. What is the maximal area of $ABC$ in terms of…
student
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How to prove Euclidean space impossibility in thesis?

There is problem where you can not unfold faces of 3D object in a way were all plane sides stay connected with each other in 2D Euclidean space. How this problem is named and how I can make reference to this impossibility in my thesis?
Timo
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Plane geometry question

Let $ABC$ be an equilateral triangle. $D$ is on $BC$ and $\angle BAD = 20^\circ$. Also let $I_1$, $I_2$ be the inner centers of triangles $ABD$, $ACD$ respectively. $E$ is a point making the triangle $I_1I_2E$ be equilateral ($D$ and $E$ are on the…
EUNSUNG LIM
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Limit of iterated midpoint polygons

Consider an arbitrary polygon $p$, and then take its midpoint polygon $p'$. Repeat this process to create $p''$, $p'''$, etc. Is there always a convex polygon in this series?
Thomas Anton
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Construct an ellipse given foci and a tangent line

Given the two foci of an ellipse and a tangent line of the ellipse, can one construct the ellipse with a compass and a straightedge?
pepa.dvorak
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Center of Mass of Quadrilateral

I recently started studying Mass Points and the question arose: If you have a quadrilateral with a mass of 1 at each vertex, how do you locate the center of mass. I had several approaches but I was not sure if they all result in the same point or…
Scott
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Construct a parallel to a line from a point outside the line in three euclidean moves

I'm trying to figure this out. It's one exercise of a set of 20, but this is the only one giving me problems. The teacher of the course mentioned that it has to be done with two circumferences (the third move is the parallel line itself). So far I…
Emmy
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How to show $\sigma_\phi \circ \sigma_\psi=\rho_{2(\phi-\psi)}$

How can I show that $$ \sigma_\phi \circ \sigma_\psi=\rho_{2(\phi-\psi)}, $$ where $\sigma_\phi$ is a reflection about a line making an angle $\phi$ with the x-axis, and $\rho_\psi$ is a rotation about the origin with angle $\psi$. Is it possible to…
Sha Vuklia
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An Application of Apollonius' Theorem

I'm working through some exercises in preparation for midterms and I'm stuck on the following exercise. For ABCD a convex quadrilateral with $AB = 13 = BD$, $BC = CD$, $AD = 10$ $\measuredangle BCD =90$ Find the length of CM where M is the midpoint…
TheCoolatron
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Choosing tangent point for incircle

In a circumscribed quadrilateral, the two pairs of opposite sides add up to the same total length: $a+c=b+d$. Conversely, any quadrilateral with $a+c=b+d$ must be circumscribed. Suppose we are given four lengths $a,b,c,d$ with $a+c=b+d$, and…
pi66
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Triangle and rational numbers

A triangle has rational side lengths and rational angles measured as degrees. Is such a triangle necessary equilateral?
user6610
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"Triangle XYZ is similar to triangle RST". Why is this considered a mathematical statement?

I am told that the statement, "Triangle XYZ is similar to triangle RST", is a mathematical statement. A mathematical statement is a statement that is either true or false. My thoughts were that this cannot be a mathematical statement, since the…
The Pointer
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find point on the line in $R^n$

I am trying to find the coordinates of a middle point of a line in $\mathbb{R}^n$. Let $X(x_1, \ldots, x_n)$ and $Y=(y_1, \ldots, y_n)$ be two points in $\mathbb{R}^n$. How do I find the middle point $Z$ on the line $XY$? Thank you.
user202312
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Hard problem of circumcircle, circumcenter and orthocenter

This question is from a list of training for math olympiad. It comes from a Brazilian olympic training called POTI. Let ABC be a triangle of a circumcircle $w_1$, $O$ be the circumcenter of ABC and $w_2$ be the excircle relative to the BC side. If…
Rafael Deiga
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line is the perpendicular bisector of a segment obtained by reflecting twice through two rays

Given two rays, L and M, with common origin O, and a point Q inside the acute angle formed by the rays, reflect Q across L to obtain Q' and then reflect Q' across M to obtain Q''. Similarly, reflect Q across M to obtain P' and then reflect P'…
samuel
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