Questions tagged [euclidean-geometry]

Geometry assuming the parallel postulate: in a plane, given a line and a point not on that line, there is exactly one line parallel to the given line through the given point.

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Rotating the Sides of a Quadrilateral Rotates its Diagonals

Rotating the 4 sides: Given ABCD, rotate AB by $\theta$ to get A'B', and rotate CD by $\theta$ to get C'D'. (When we speak of rotating a segment by $\theta$, we mean rotating it by an angle of $\...
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Straightedge-and-compass construction of the “kissing circles” for three given circles

Let $C_1,C_2,C_3$ be three mutually tangent circles. Call the circle tangent to all of them (that is, intersecting each at one point) and enclosed within the region between them their kissing circle. ...
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Right tangential trapezoid

In a right tangential trapezoid $ABCD$ $(AB\parallel CD)$ and $AD\perp AB$ the incircle is $k(O).$ Find the area of the trapezoid if $CO=6$ and $BO=8.$ The triangle $BOC$ is a right triangle and by ...
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Construct a circle tangent to sides $BC$ and $CD$ and s.t. its meetings with the diagonal $BD$ are tangent points from tangents draw from point $A$

Given square $ABCD$ I want to construct (with ruler and compass) the circle in the interior of the square such that it is tangent to sides $BC$ and $CD$ and such that it's meetings with the diagonal $...
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Recommend book on elementary geometry

I am seeking recommendations for books on elementary geometry, including Euclidean geometry and analytic geometry.
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The altitude of an isosceles trapezoid; given area and diagonal

An isosceles trapezoid is given with diagonal $25$ $cm$ and area $300$ $cm^2.$ Find the altitude and the midsegment. Let $DD_1$ and $CC_1$ be the altitudes of the trapezoid through $D$ and $C,$ ...
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Is there a citation or (at least a date) to ascribe to Conway's Circle Theorem?

There are lots of John Conway tributes (via cardcolm.org) appearing online, and one of the things that I see a lot is the fact that his Circle Theorem (via MathWorld) is a geometric construction that ...
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1answer
29 views

Ratio of the heights of triangle is given, determine the sides of triangle.

Can someone help me with this excersize? The ratio of heights to the sides of the triangle is v_{a}:v_{b}:v_{c}=12:5:8. What are the length of the sides of triangle? (a,b,c=?) Thank you! I tried ...
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54 views

$3(a+{1\over a}) = 4(b+{1\over b}) = 5(c+{1\over c})$ and $ab+bc+ca=1$

This was the question : $3(a+{1\over a}) = 4(b+{1\over b}) = 5(c+{1\over c})$ where $a,b,c$ are positive number and $ab+bc+ca=1$ , $5({1-a^2\over 1+a^2}+{1-b^2\over 1+b^2}+{1-c^2\over 1+c^2}) = ?$ ...
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1answer
27 views

How to get the left and right rotation quaternions between two $\mathbb{R}^4$ unit vectors?

This question is somewhat similar to this, but in higher dimension. I have two non-collinear unit vectors in $\mathbb{R}^4$, $\bf a$ and $\bf b$. I know from Wikipedia that there is at least one pair ...
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Trisecting an angle with a compass and 2 marks on a ruler

It's well known that there is no possibility to trisect and angle with a compass and a ruler. But there is such a procedure when the ruler has 2 marks on it. See the snippet below. It's not very ...
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Double/Volume integral in parametric form

Find the volume of the tetrahedron with vertices at $(0,0,0),(a,0,0),(0,b,0),(0,0,c)$ The most straight forward approach would be evaluating the following integral in Cartesian coordinates system. $$...
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Minimum circumsribed square

I want to find the minimum circumsribed square of the regular polygon with 2n edge. n is the odd number.I have known the method by observing some simple condition. But how to prove it strictly.
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Finding angle between two hyperplanes given basis vectors

There is a Euclidean space $\mathbb{R}^n$. In it there live two hyperplanes of dimension $m$ each. Both hyperplanes pass through the origin $\vec{0}$. The hyperplanes are defined by the corresponding ...
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Quadrilateral with two congruent legs of diagonals

I've come across a geometry proof which seems like it should be easy, but I'm struggling with it: Suppose you have a convex quadrilateral $ABCD$ whose diagonals intersect at $E$. Given that angles $...
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angles and points interior and exterior to them

I came upon the following exercise. It seems so obvious, yet I don't know on which axiom (or more fundamental theorem) to base the proof. Let $xOz$ be an angle, let $A$ be a random point on the ...
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Is this geometric proof logically circular?

Here is the problem: Prove that in a triangle $ABC$ with $C$ as the right angle, where $a$ denote the side in front of angle $A$, $b$ denote the side in front of angle $B$, $c$ denote the side in ...
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Is there an explanation/theorem which explains why the triangle is so fundamental in euclidean space?

I was just thinking the other day: a triangle is the simplest shape in euclidean space and so it seems obvious why all other shapes in this space could be built form it. It is kinda like the atoms of ...
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80 views

The area of an isosceles trapezoid; given midsegment, diagonal and leg

The midsegment of an isosceles trapezoid is $4$, the diagonal is $4\sqrt{2}$ and the leg is $2\sqrt{5}$. Find the area and the bases. Let $DD_1$ and $CC_1$ be the altitudes and $MN$ be the ...
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Find the area of the entire shape.

ABCD is a straight line. ABE is a sector of a circle with center B. CED is a sector of a circle with center C. Angle ABE is a right angle. The length of AB is r and angle ACE is π/4 radians. If r=10cm,...
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On non-canonical Euclidean metric on $\mathbb{R}^n$

The canonical Euclidean metric on $\mathbb{R}^n$ is as $\sum_{i=1}^ndx_i^2$. We also know that every Riemannian space of zero sectional curvature is Euclidean. Therefore, for example, $\sum_{i=1}^...
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I need an explanation of the end of the problem [closed]

All I need is to know how they get to pi / 6 (what's in the circle) note:sen x =sin x note: it is a single problem that uses trigonometry along with its development [problem][2] ...
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Strange Concurrence on lines.

I got really stuck proving this exercice of my text on geometry. Can anyone provide a solution please? If the lines $AA'$, $BB'$, $CC'$ concur, then the points of intersection of the lines: $AB,A'B'$;...
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31 views

Cevians concuring and a sum.

Geometry problems that me and my squad cannot solve. This quarantine, we decided to make some geometrical research, we got stucked on this: In a $\triangle ABC$ triangle, let the Cevians $AL$, $BM$, $...
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Find the inradius in a tangential quadrilateral

In this tangential quadrilateral $AD+BC=22$, $AB=9$, $CF=8$ and $F$ is a tangent point. Find the inradius $r$. Here is my attemps: Using Pitot's theorem I found that $x+y=9$. Also, using the area ...
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1answer
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A property of trapezoids - Steiner's Theorem

An important feature of trapezoids is that the midpoints of its bases, the intersection of its diagonals, and the intersection of the lines through its legs are collinear. Explanation for three of ...
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Properties of reflections in Euclidean space

I am seeking a deeper understanding of Euclidean space(es). In the Euclidean plane, the lines are of major interest. When using a line to reflect points, lines are mapped to lines. In higher ...
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Proof by using Euclid's Elements book

i hope all you are fine. I have a question from Euclid's Element's book IX-14. It says that "If a number is the least that is measured by prime numbers, then it is not measured by any other prime ...
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Find $\epsilon$ of a finite size $\epsilon$-net of a $d$-dimensional unit ball$

I have a set of $d$-dimensional vectors $V = \{+1, 0, -1\}^d $. Then $P(V)$ constitutes the power set of $V$. I now construct a set of unit vectors $V_{sum}$ from the power set $P(V)$ such that $$ V_{...
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The base of an isosceles triangle; given leg and radius of circumcircle

An isosceles triangle $\triangle ABC$ is given with leg $AC=5$ and $R=\dfrac{25}{6}$ of the circumcircle. Find the base of the triangle. This was my first sketch. The triangles $AHC$ and $OMC$ are ...
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Express a vector in terms of a and b and find the angle

The following figure shows an equilateral triangle OAB with side length h. C is the mid-point of AB. D is a point on OA such that OD:DA = 2:1. OC intersects BD at E. OBFE forms a parallelogram. Let ...
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1answer
41 views

The leg of an isosceles triangle; given base and area

The area of an isosceles triangle $\triangle ABC$ with base $AB=2c$ is $S.$ Find the leg of the triangle. Let $AC=BC$ and $CH$ and $BP$ be the altitudes through $C$ and $B$, respectively. I am not ...
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Inscribe a triangle in the given triangle

In a given triangle ABC inscribe the triangle A'B'C' whose sides are parallel with three given lines p ,q ,r ( lines are given as sides of a triangle).
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Prove that $BQ$ bisects segment $EF$.

Question: Let $ABC$ be an acute angled triangle with orthocentre $H$ and circumcircle $\Gamma$. Let $BH$ intersect $AC$ at $E$, and let $CH$ intersect $AB$ at $F$. Let $AH$ intersect $\Gamma$ again at ...
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Is it possible to prove the midpoint theorem with just using alternate-interior(exterior) or corresponding angles?

Here is the statement : Let $ABC$ a triangle, $I$ is the midpoint of $[AC]$ and $J$ is the midpoint of $[BC]$. Then the lines $(IJ)$ and $(BC)$ are parallels and $2IJ = BC$. So is it possible to ...
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Pythagorean theorem by contradiction

This proof cannot be found in cut-the-knot.org nor in Loomis' collection. Let $\triangle{ABC}$ be a right-triangle with $\angle{ACB}=90^\circ$. Let $D$, $E$ and $F$ be the contact points of the ...
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The base of an isosceles triangle; given the radius of the inscribed circle and perimeter

On the sketch below $AC=CB$ and $OD=\dfrac{4}{10}CD$. If the perimeter of $\triangle ABC$ is $P_{\triangle ABC}=40$, find the length of the base $AB$. Let $AC=BC=a$ and $AB=c$. Since the triangle is ...
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How to verify that matrix is a correct Euclidean distance matrix in $k$-d space?

Given symmetric non-negative matrix $D \in \mathbb{R}^{n\times n}$, how to verify that we can place points $x_1,\ldots x_n$ in $k$-d space such that $D_{ij}=\|x_i - x_j\|_2$? If there is no general ...
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82 views

Does connecting any two points in a graph result in a convex set? [closed]

This is a follow-up question. Let $F:[0,1] \to [0,\infty)$ be a continuous function, and let $G=\{ (x,F(x))\,|\, x \in [0,1] \}$ be the graph of $F$. Is $\cup_{(x,y)\in G^2} [x,y]$ convex?
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Does connecting any two points in a set result in a convex set?

This might be silly, but I am not sure. Let $A \subseteq \mathbb R^2$. Suppose that for any two points $x,y \in A$, I "add" the straight segment $[x,y]$ between them. Is the result convex? That is, ...
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Confusion constructing the triangle $ABC$ given the lengths $AB$, $AC$ and the length of the median $m_a$?

I'm trying to answer the following problem: I did in a way and went to look for the solution, this is it: I tried to follow it using Geogebra, I think it goes this way: Given the setup of ...
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The points $M$ and $N$ are chosen on the angle bisector $AL$ of a $\Delta ABC$ such that $\angle ABM=\angle ACN=23^0$.

Question: The points $M$ and $N$ are chosen on the angle bisector $AL$ of a $\Delta ABC$ such that $\angle ABM=\angle ACN=23^0$. $X$ is a point inside the triangle such that $BX=CX$ and $\angle BXC = ...
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How do I approach finding the radius with the measurements given?

A parallelogram with three congruent inscribed circles is painted across a 20-foot by 12-foot rectangular banner as shown. The longer sides of the parallelogram have length 15 feet. What is the radius ...
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53 views

The perimeter of an isosceles triangle $\triangle ABC$

An isosceles triangle $\triangle ABC$ is given with $\angle ACB=30^\circ$ and leg $BC=16$ $cm$. Find the perimeter of $\triangle ABC$. We have two cases, right? When 1) $AC=BC=16$ and 2) $AB=BC=16$. ...
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1answer
23 views

$\Lambda, \Theta$ subspaces in $\mathbb{E^n}$

Let $\Lambda, \Theta$ be two subspaces in $\mathbb{E^n}$ such that $\Lambda, \Theta$ are skew and $dim(\Lambda )+dim(\Theta)=n-1$. If we call $\Psi$ the hyperplane such that $\Lambda\subseteq \Psi$ ...
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Geo question about lengths [closed]

I have a question from a maths paper. Let $ABC$ be an isosceles triangle with base $BC$ and Angle $BAC$ = 100. The bisector of Angle $ABC$ intersects $AC$ in $P$ . Show that $BC$ = $AP$ +$PB$. ...
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40 views

In what order are the points on the hypotenuse?

$\triangle ABC$ is a right triangle with $\angle ACB=90^\circ$. Let $CM=m_c$ be the median and $CH=h_c$ be the altitude through $C$. I want to understand how I should approach problems with ...
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1answer
40 views

Doubt for the area of shaded region for trapezium

I wonder if I can use the area of trapezium for the shape CAOB? I've abit of doubt for the arc slope as often, the slope of the trapezium appears to be a straight line. For part a) I'm just checking ...
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2answers
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Comparing interior angles and dihedral angles in tetrahedra

Let $S\subset\Bbb R^3$ be a tetrahedron (not necessarily regular, just the convex hull of any four points in general position). Let $v,e,\sigma\subset S$ be a vertex, an edge and a face of $S$, so ...
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1answer
58 views

Can an angle between the subdividing segments and the edges of a triangle be determined only by interior angles and the intersection of the segments?

I'm working on a robotics project, where a body $(D)$ is to be linked to three anchor points $(A, B, C)$ with segments of variable length. The movement envelope is the area of the $\triangle ABC$, ...

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