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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If we have the slope of $AB$ and $AC$. How can we determine the angle of $AB$ and $AC$?

If we have the slope of $AB$ and $AC$. How can we determine the angle of $AB$ and $AC$? I searched the internet but I don’t understand. Please help! Thank you very much.
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Given a random point and a cube, how can I determine the distance from that point to the furthest possible point on the cube?

Given a random point in space ( $\vec{p}$ ), I am trying to figure out how to calculate the distance to the furthest point on an axis-aligned cube/cuboid from that other point. This cube is defined by ...
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Prove that $d^2=a^2+b^2+c^2-2[ab\cos B+bc\cos C+ac\cos(α)]$ [closed]

The angle between the AB and CD sides of an ABCD convex quadrilateral is equal to $\alpha$. Considering that AB = a, BC = b, CD = c, DA = d, prove that: $$d^2=a^2+b^2+c^2-2[ab\cos B+bc\cos C+ac \cos(\...
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20 views

Prove that $MK=ML$

In a right triangle ABC such that $\angle C=90°$, denote $CD$ be the height of the triangle. Let $X$ be any point on the segment $CD$ then let $K,L$ be the intersection of $AX$ and $BK$, $BX$ and $AL$ ...
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Moscow Maths Olympiad 1952, geometry problem [duplicate]

In an isosceles triangle $ABC$ , $AB=BC$, $\angle B=20$ degrees. $M, N$ are on $AB$ and $AC$ respectively such that $\angle MCA=60$ degrees and $\angle NAC=50 $ degrees. Find $\angle NMC$ in degrees. ...
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24 views

Problem on an isosceles right triangle, involving similarity and congruence

Given that $ABC$ is an isosceles right angled triangle with angle $\widehat{ACB}=90$ degrees. $D$ is the midpoint of $BC$, $CE$ is perpendicular to $AD$, intersecting $AB$ and $AD$ at $E$ and $F$ ...
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61 views

Radius of circle that touches 3 circles, which in turn touch each other

I had $3$ circles of radii $1$, $2$, $3$, all touching each other. A smaller circle was constructed such that it touched all the $3$ circles. What is the radius of the smaller circle? This is what I ...
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46 views

Proving the converse of Ceva's theorem

This is what the converse of Ceva's theorem states. Suppose $ABC$ is a triangle, and let $AD$, $BE$, $CF$ be the three cevians such that $$\dfrac{BD}{DC}\cdot\dfrac{CE}{EA}\cdot\dfrac{AF}{FC}=1$$ ...
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33 views

Equilateral triangle in a regular pentagon

I don't understand how the line segment of $|BF|$ equals to $|BC| = |FC| = |AB| = |AE|$. How can the line segment maintains the equilateral triangle? The instructor says while drawing that it just ...
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Intuition behind area of ellipse

This is not meant to be a formal proof, but I just wanted to know if this is a valid way of thinking about the area of an ellipse. It does assume knowledge of the area of a circle, but this can be ...
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52 views

Find a certain invariant angle, given a constraint on perimeter

Each side of square ABCD has length 1 unit. Points P and Q are on AB, and DA, respectively. Find angle PCQ if the perimeter of triangle APQ is 2. I used trigonometry to solve this problem, and my ...
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26 views

covering a rectangle with an ellipse of minimal area

Given a rectangle $B$ I would like to find an ellipse $C$ such that $B$ is contained inside $C$. Of course this is always possible, but I was wondering can we do this in a way that the area of $C$ is ...
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27 views

Prove that,a triangle can be formed using sides equal to the diagonals of any convex pentagon.

A friend of mine recently gave me a problem saying, $\textbf{Question:}$ Take any convex pentagon.Show that, we can make a triangle whose side lengths will be distinct diagonals of the pentagon. I ...
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26 views

Locus of points and angles

The angle bisector at A of triangle ABC cuts BC at L. If C describes a circle whose center is A and B remains fixed, what is the locus of the points L? I tried some sketches and some geogebra but I ...
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A rare property on Complete Quadrangles

Studying complete quadrangulars I found an interesting question that I cannot solve: Can the diagonal points of a complete quadrangular be collinear? ABCD is a complete quadrangle And the ...
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1answer
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Find the height of a tetrahedron, given the length of all base edges and angle of lateral faces

Find the volume of a triangular pyramid which has base with edges of length 4, 5 and 7, and lateral faces with the base form a 48 degrees and 30 minutes angle. Solution is volume $V=4.52$; base $B=4\...
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24 views

Relation between areas in a trapezoyd

A trapezoid of ABCD vertices is inscribed in a circle, with radius R, being AB = R and CD = 2R and BC and AD being non-parallel sides. The bisectors of the internal trapezoidal angles, so that the ...
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1answer
45 views

Length of a line in a regular hexagon

The picture belongs to a regular hexagon. I don't understand that if a line is drawn from $E$ to $C$, how its length equals to $|KL|$ and how it is parallel to $|KL|$? Would you mind drawing its ...
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How to transport a segment to a line

The exercise is 2.11 of Hartshorne Euclid And Beyond, and it says: Given a segment d, a line l and a point O, construct a circle with center O that cuts off a segment congruent to d on line l. (9 ...
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33 views

Which point on a given line maximizes anglular separation of two given points?

Suppose points $F_1$ and $F_2$ are the left and right foci of the ellipse $(x^2/16)+(y^2/4)=1$, respectively and point $P$ is on the line $L$ which is $x-\sqrt3y+8+2\sqrt3=0$. When $\angle F_1PF_2$ ...
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36 views

In triangle ABC, diameter and radius of circumcircle meet BC at Q and M. Prove $\frac{[AQC]}{[MTC]}=\left(\frac{\sin B}{\cos C}\right)^2$

In triangle ABC, the height from A cuts the circumcircle at T. The diameter of the circle that passes through A and the radius OT cut the side BC at Q and M, respectively. Demonstrate that $$\frac{[...
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Total distinctive nets of regular icosahedron (20-face regular polyhedra)

Came across this on wikipedia: An icosahedron has 43,380 distinct nets. But I can't find the proof to it. Is there an easy to understand proof? Also, if I color the each face of icosahedron with ...
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28 views

Finding the ratio between the area of a circle inscribed by a kite and a circle inscribing the kite

In the following problem, $\angle DAB = 2\alpha$, and $ABCD$ is a kite ($AD=AB, DC=CB$). I need to prove the ratio between the circle inscribed by the kite to the area of the circle inscribing the ...
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40 views

How many points {0,1}^k can a hyper plane in n-dimension contain?

Suppose we have a hyperplane H in $\mathbb{R}^n$ and a set of points S = $\{0,1\}^k,\; k\leq n $. We want to find the maximum number of points that the hyperplane H can contain from the set S. edit: ...
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122 views

Find an angle in the given quadrilateral

In the following problem, I want to find the angle marked as $x$. It seems so simple and yet I am out of ideas. It is very easy to get all angles except two of them: angle ADB and angle CBD. Is ...
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1answer
39 views

Finding the circumradius-to-inradius ratio for the regular pentagon

Find the value of $R/r$: I go with co-ordinate geometry, considering the centre of the circles is at the origin, then the equation of the circle becomes as $$ x^2 + y^2 = R^2 $$ $$ x^2 + y^2 = r^2 $$...
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2answers
49 views

Triangles and inequalities

Let ABC be a triangle and let O be any point in space. How can I show that $AB ^ 2 + BC ^ 2 + CA ^ 2 \leq 3 (OA ^ 2 + OB ^ 2 + OC ^ 2)$? I know this prove by inner products, but is possible to show ...
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1answer
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Are left isoclinic rotations a group?

Assume the following definitions: Isoclinic rotations are rotations $\varphi$ in $\mathbb{R}^{2n}$ such that there exists $n$ complementary oriented planes $P_i=\langle x_i,y_i\rangle$ such that $\...
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How to solve without using trig?

In the diagram, five identical squares have been placed together. What is $\angle ABC$? It's easy with trig but can't find an answer without using it. Thanks!
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Triangle with altitudes and projections.

Let $ABC$ be triangle with $AA_1$ , $BB_1$ and $CC_1$ as its altitude. $M$ and $N$ are the projection of $C_1$ to $AC$ , $BC$ respectively. Let $MN$ intersects $B_1C_1$ at $P$. Show that $P$ is ...
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1answer
30 views

A minimum movement of a sphere to avoid superposition from another

Let's say there are 2 sphere $S_1$, $S_2$ with same radius $k$, where the centers of them are $C_1:(x_1,y_1,z_1)$, $C_2:(x_2,y_2,z_2)$, respectively. If there exists superposition, where the ...
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1answer
46 views

Equations of the orthogonal projection on the line $r = y - 2x + 1 = 0$

I'm trying to calculate in the Euclidean affine plane with respect to an orthonormal reference the equations of the orthogonal projection on the line r of equation y - 2x + 1 = 0. So, let $\phantom{3}...
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2answers
94 views

Triangles and medians

Let G be the ABC barycenter. A line intersects the medians AD, BE and CF in X, Y and Z, respectively. Prove that $$\frac{XD}{XG}+\frac{YE}{YG}+\frac{ZF}{ZG}=3$$ By areas relations I found that $\frac{...
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1answer
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Can the cross section of parallelepiped be a regular pentagon

Came across this question in a children's recreational mathematics book. Apparently, the cross section of a cube cannot be a regular pentagon. It could be a irregular pentagon though. But if we ...
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On circumcircle, incircle, trillium theorem, power of a point and additional constructions in $\triangle ABC$

The problem was at this deleted question originally. Given: 1) $\triangle ABC$ -- an arbitrary triangle 2) with circumcircle $\omega$ centered at $O$ 3) and incenter $I$. 4) Let $D$ be the ...
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Basic Geometry Problem : FInd $AC$

Let $ABC$ be a triangle where $\angle B$ is a right angle. Extend $AC$ up to point $D$ such that $\angle CBD=30^\circ$. If $AB=CD=1$, find $AC$. My approach to this problem is first to try to find ...
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A new(?) proof of the Law of Cosines using segments determined by the points of contact with a triangle's incircle

In trigonometry, the law of cosines (also known as the cosine formula, cosine rule, or al-Kashi's theorem) relates the lengths of the sides of a triangle to the cosine of one of its angles. Using ...
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1answer
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Finding length of some sides on a triangle construction.

I must find the length of $DE$ and $EC$ Pythagoras told me the length of $AB$, and I know that the right angles give me similar triangles, but I don't know how to proceed. Also tried Pythagoras on ...
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88 views

Prove that $\frac{PD}{DM} = \frac{BP}{BM}$

Given a quadrilateral $ABCD$, lines $AD,BC$ meet at $R$ and lines $AB,CD$ meet at $Q$. The diagonals $AC,BD$ meet at $P$. Also, let $M$ be the intersection of $BD$ and $QR$. Prove that $\frac{BP}{BM} =...
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1answer
31 views

Find the function does describe the the percentage of the area that each circle overlaps

I saw this question, yesterday and it got me thinking, what function does describe the the percentage of the area that each circle overlaps. In that diagram it is given that the distance between the ...
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31 views

Locus of a moving point, when constraints on an angle and length are given

$APQ$ is a variable triangle; $A$ is fixed, $P$ moves on a fixed line $CD$; if $AP$ meets a fixed line parallel to $CD$ at $R$, and if $PQ=AR$ and if the angle $APQ$ is constant, prove that the locus ...
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1answer
39 views

Locus of a moving point, such that two distances have a common ratio

A, B are two fixed points on a fixed circle; P is a variable point on the circle; Q is a point on BP, such that BQ/AP is constant; find the locus of Q. The only approach I could think of is through ...
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1answer
19 views

Relation between the radius of $n$ identical circles and the radius of an enclosing tangent circle

$n$ small circles are tangent to each other and tangent to the big circle. Here's a figure for $n=4$: Asking hints of how to find the reason between the radius of small circles into the big ...
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1answer
48 views

How to find a point which is outside a bunch of planes in 3D?

I would like to find an arbitrary $\mathbf{p} \in \mathbb{R}^3$ point which is not included in any of planes defined by surface normals $\mathbf{s}_1, \mathbf{s}_2,..., \mathbf{s}_m \in \mathbb{R}^3$ (...
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2answers
32 views

Existence of $f$ such that $f(x,|x|^2)f(y,|y|^2)=0$ whenever $x \cdot y=0$

Does there exists a non trivial continuous function (other than $f=0$) with the following : $f:R^4 \to [0, \infty)$ Let a $x,y \in R^3$ and their respective Euclidean norm squared $|x|^2$ and $|y|^...
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geometry problem dealing with quadrilateral and proving two lines perpendicular to each other and equal in magnitude

On each side of a quadrilateral ABCD,squares are drawn.The centers of the opposite squares are joined.Show that PR and QS are perpendicular to each other and equal in magnitude. pure geometry is ...
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36 views

Euclidean geometry- how do I finish this proof

The problem at hand: In triangle $ABC$, $D$,$E$,$F$ are points on the sides BC,CA,AB. A,B,C are further points on a bigger triangle $XYZ$ such that $EF$ $||$ $YZ$, $FD$ $||$$ZX$, $DE$ $||$ $XY$ Prove ...
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1answer
50 views

Derivative of double centered Euclidean distance matrix

I want to do this matrix calculus: Given, a distance matrix of squared Euclidean distances $D(X)_{n \times n}$ of $n$ points $X \in \mathbb{R}^k$, and given $C_n$, a centering matrix as defined in ...
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2answers
107 views

Really really hard and old Euclidean geometry problem

Let $M$ be the midpoint of side $AB$ of triangle $ABC$. Let $P$ be a point on $AB$ between $A$ and $M$, and let $MD$ be drawn parallel to $PC$ and intersecting $BC$ at $D$. If the ratio of the area of ...
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0answers
35 views

Distance between points maximally distributed on n-dimensional unit sphere?

This problem arose in some of my own personal data science research and I am wondering if anyone has encountered this before. Consider $k$ points that lie on an $n$-dimensional unit sphere such that ...