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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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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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 ...
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Is there a unified formula for the lengths of the median, angle bisector, and altitude of a triangle?

The (squares of the) lengths of the median, angle bisector, and altitude of an $a$-$b$-$c$ triangle with base $a$ are given by the following: $$\begin{align} d_{\text{med}}^2 &= \frac14\left( - a^...
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How to prove an angle equals to $90^°$ algebraicly

Let $ABCD$ be an isosceles trapezoid. Let $P$ be the intersection of its diagonals. the line $DA$ meets the circumcircle of $APB$ at $M$. Let $L$ be a point on $AB$ such that $LD=LB$ and $K$ be a ...
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Proof of co-ordinates of a point which is nearest to atleast K points

I think this is related to K-nearest neighbors however I have not been able to wrap my head around it. First the problem, we need to prove the following, Given co-ordinates of $N$ points, $(x_i, ...
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3answers
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Do Tarski's (geometry) axioms imply that all zero segments are congruent?

Tarski's axioms are an alternate formalization of geometry (similar to axiom sets of Euclid and later Hilbert). Do these axioms imply: $$\forall\; x,y\in \text{points},\; x x\equiv y y?$$ If yes, ...
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Nice geometry problem involving two touching circle

$\textbf{Question:}$ . Circle $O_1$ and $O_2$ touches each other externally at a point T , quadrilateral ABCD is inscribed in $O_1$, and the lines DA, CB are tangent to $O_2$ at points E and F ...
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tetrahedron $OABC$ of edges $a,b,c$.Let $G_1 $, $G_2 $ be centroids of $ABC $, $AOC $ such that $ OG_1\bot BG_2$, prove $a^2+c^2=3b^2.$ [closed]

In a tetrahedron $OABC$, the edges are of lengths, $|OA|=|BC|=a, |OB|=|AC|=b, |OC|=|AB|=c. $Let $G_1 $ and $G_2 $ be the centroids of the triangle $ABC $ and $AOC $ such that $ OG_1\bot BG_2,$ then ...
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What is the radius of the small circle inscribed in a square?

If we are given a square with sides of length 4cm. The smaller circle is tangent to the larger circle and the two sides of the square as shown in the photo below. How can i find the length of the ...
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Bisectors and Incenter of a triangle

Let G be the barycenter, I, the incenter, R the circunradius, r the inradius and p the semiperimeter of a triangle. Prove that $GI^2=\frac{p^2+5r^2-16Rr}{9}$ I did this question using analytical ...
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What is the rule of the dilation?

I was able to find the center of enlargement $(3,0)$ and the scale factor $-\frac{1}{2}$. I tried to write the rule for the dilation and I had $(x,y) \rightarrow (-\frac{1}{2}x+\frac{9}{2}, \frac{1}{2}...
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1answer
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Angle between two midpoints equals angle between point at perpendicular from top to base to other midpoint

Let, X, Y, Z be the midpoints of the sides AB, BC, CA of the triangle ABC. Let P be defined on BC so that ∠CPZ = ∠YXZ. Prove that AP is perpendicular to BC. This question is from a book I'm reading ...
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How to divide a unit space into many simplexes?

I'm sorry, it may be simple and stupid but I didn't find any relative solutions on the Internet. Given the unit hypercube $C$ in the Euclidean space $R^n$, how to divide (or we can say "partition", I ...
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1answer
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A parallel line through the incenter of a triangle

The circumcircle of $\triangle ABC$ is $k(O;R).$ Through the incenter of the triangle is drawn a line $p$ parallel to $AC$ that intersects the circle $k$ at $M$ and $N.$ The side $AB=30$ divides the ...
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“Heron's proof of AL,BK.CF in Euclid's Figure meet in a point” in God created the Integers.

I was going through the section "Heron's proof of AL,BK.CF in Euclid's Figure meet in a point", in God Created the Integer's book . The statement of problem and its proof are both in very vague form. ...
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1answer
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Triangles and circles

How can I generalize this problem (Four circles tangent to each other and an equilateral triangle)? I mean how can I express R as a function of $R_1,R_2,R_3$ in this triangle: I applied a lot of ...
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Prove equality of angles in a square [closed]

Let ABCD be a square. Choose points $E \in BC$ and $F \in DC$ such that BE + DF = AE. Prove that $\angle DAF$ = $\angle FAE$ (Some kind of converse of Equality of segments in a square)
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When do three closed balls have a nonempty intersection?

Consider a real Hilbert space $\mathcal{X}$. For $(c,\rho)\in\mathcal{X}\times \mathbb{R}$, I denote the closed ball $B(c;\rho) = \{x \in \mathcal{X}|\|x -c\|\leq\rho\}$. I am curious if y'all know of ...
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2answers
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Compare time of day using Euclidean distance

I have $t_1, t_2, ..., t_N$, where each $t \in [0, 1[$ is a time during the day (i.e. 0.01 is right after midnight and 0.99 is right before midnight). I want to compute the distance between these time ...
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1answer
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Intersecting Secants Theorem

Let the point $A$ lie on the exterior of the circle $k(R).$ From $A$ are drawn the tangents $AB$ and $AC$ to $k.$ The triangle $ABC$ is еquilateral. Find the side of $\triangle ABC$. I am not sure ...
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Average shortest path and diameter in Poisson-Delaunay graphs

For a given set of $N$ random points, distributed uniformly on a unit square, I construct its Delaunay triangulation. Taking the triangulation as an unweighted graph, I need to know the expected ...
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1answer
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Prove that orthocenter of the triangle formed by the arc midpoints of triangle ABC is the incenter of ABC

Let $ABC$ be an acute triangle inscribed in circle $W$. Let $X$ be the midpoint of the arc $BC$ not containing $A$ and define $Y$, $Z$ similarly. Show that the orthocenter of $XYZ$ is the incenter $I$ ...
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Calculus: Early Transcendentals, 7th ed(stewart)-chapter 12 problems plus exercise 3

I tried to resolve the following excercise but i got stuck. 3)Let be $L$ the line of intersection of the planes $cx+y+z=c$ and $x-cy+cz=-1$, where $c$ is a real number. (a) Find symmetric equations ...
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Are the two triangles similar?

A book I was answering was asking for the center of enlargement and the scale factor of the two similar figures.But I don't think the two triangles are similar. They corresponding sides don't have the ...
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1answer
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Right circular cone shortest distance question from a junior high school st. [duplicate]

Ant's shortest distance that it can travel Can you show me the answer in the cone unfolded. I would be glad if you guys can show me the solution in a simple way because I am a high school student.
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Prove that the area of the triangle formed by the centers of the circumferences ex-inscribed is $\frac{abc}{2r}$

Prove that the area of the triangle formed by the centers of the circumferences ex-inscribed is $\frac{abc}{2r}$, where $a,b,c,$ are the sides of the triangle and $r$ is the inradius. I know that $S=...
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Solving the integral with norm vector over bounded region

I have to solve an integral like this: $$ d = \iint\limits_R{||x||^{-2/3} dA} $$ And the problem I have is described like this: *there is a disk region R having an area of 100 m² and in this ...
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1answer
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Sine difference identity

I'm trying to prove that given 2 vectors $\vec{a} = A(\cos{\alpha}, \sin{\alpha}) $ and $\vec{b} = B(\cos{\beta}, \sin{\beta}) $ the following relation is true by using the exterior product with the ...
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3answers
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Two inscribed circles are tangential to a chord with diameter line at $30^\circ$ with chord. Find radius ratio of two circles

I found the problem here (I can't see deleted posts) but the post got downvoted and deleted soon, but I felt so inspired to find the solution that can't let the problem rot by itself, so, rather re-...
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Difficult problem in elementary euclidean geometry

A point D is chosen inside an equilateral triangle $ABC$ such that $AD$ = $BD$. A point $E$ outside the triangle is chosen such that $\angle DBE$ = $\angle DBC$ and $BE$ = $AB$. Find the degree ...
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1answer
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Can I make this assumption?

I am solving this question: Let $ABC$ be an acute angled triangle and $CD$ be the altitude through $C$. If $AB=8$ and $CD=6$, find the distance between the midpoints of BC and AD. So I observed that ...
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1answer
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Construct any regular polygon that has the same area as the sum of $n$ given triangles

Original question: Construct any regular (or similar-scaled to a given) geometric shape that has the same area as given triangle? My idea is application of generalized Pythagora's theorem. Euclid ...
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3answers
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Prove that $∡ADI=90°$

Let $ABC$ be a scalene triangle. $I$ is incenter. Common point of inscribed circle and $BC$ is $E$. $AF$ is angle bisector. If circumcircles of $ABC$ and $AEF$ meet at $A$ and $D$, then prove $∡ADI=90°...
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Property of Parallel transport in Manifold

Let M be an Hadamard Manifolds, $C$ a nonempty, closed geodesic convex subset of $M$, $T_xM$ the tangent space of $M$ at $x \in M$ and $TM$ the tangent bundle of $M$. Let $F : C \to TM$ be a smooth ...
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Geometry and tangent-chord theorem problem?

According to the figure, CA is tangent to the circle, centre O, at A. ABT and POT are straight lines. Question: Given that BT is equal to the radius of the circle, prove that: $\angle ABP = 3 \...
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Generalizing Routh's Theorem to quadrilaterals

Routh's theorem gives the area of a triangle determined by three cevians in a parent triangle, in terms of the "Ceva ratios" for those cevians. After learning about Routh's theorem, I started ...
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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 $...
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Unsure of a proof shorcut in a simple chasing the angle problem

I was given a simple geometry problem. You need to find the angle ADB. Line DB touches the circle. The angle can be easily found by making use of properties of circle and interior angles taught in ...
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Prove that A(FKP)=A(EDP).

ABCD is circumcircled in W. It is trapezoid. Mid points of parallel sides, BC and AD are E and F respectively. AB and DC intersect at P, W(circle) and BF intersect at K. Then prove A(FKP)=A(EDP) Where ...
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+50

How many whole rectangles can you catch in a grid?

There are $n$ rectangles packed in a square; all of them are axes-parallel. You are allowed to partition the square into a grid of cells, with $1$ or more rows and $1$ or more columns. You score a ...
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1answer
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Angle chasing geometry problem [duplicate]

Problem: In a triangle ABC, AC=BC and $ \angle C =20°$. There are points M and N on AC and BC respectively such that $\angle BAN=50°$ and $\angle MBA =60°$.Find the angle BMN? I tried angle chasing ...
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Show that $KL$ is parallel to $BB_1$.

The incircle of a triangle $ABC$ has center $I$ and touches $AB , BC , CA$ at $C_1 , A_1 , B_1$ respectively. Let $BI$ intersects $AC$ at $L$ and let $B_1I$ intersects $A_1C_1$ at K. Show that $KL$ is ...
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Geometry problem involving a cyclic quadrilateral and power of a point theorem?

Convex cyclic quadrilateral $ABCD$ are inscribed in circle $O$. $AB,CD$ intersect at $E$, $AD,BC$ intersect at $F$. Diagonals $AC, BD$ intersect at $X$. $M$ is midpoint of $EF$. $Y$ is midpoint of $XM$...
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Euclid's proof of Side-Angle-Side equality

I was looking at Euclid's proof of the triangle Side-Angle-Side equality, and I'm not really convinced by it. The first step of the proof is to place the first triangle onto the second one, but why ...
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2answers
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Too many tangents

Let $C_1$ and $C_2$ be two circles of unequal radius. Circles $C_1$ and $C_2$ intersect at points $A$ and $B$; let $L_1$ be the tangent line to $C_1$ at $A$, and let $L_2$ be the tangent line to $C_2$ ...
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1answer
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Find the sides of a parallelogram

A parallelogram $ABCD$ is given. Let $DP$ be perpendicular to the diagonal $AC$ $(P\in AC).$ If $AP=6$ $cm$ and $CP=15$ $cm$ and the difference between the sides of $ABCD$ is $7$ $cm,$ find $BD.$ If ...
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1answer
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Cevianas AD, BE and CF of ABC compete at P. Show $\frac{S_{DEF}}{2S_{ABC}}=\frac{PD .PE.PF}{PA.PB.PC}$

In an ABC triangle, the cevianas AD, BE and CF compete in P. Show that $\frac{S_{DEF}}{2S_{ABC}}=\frac{PD .PE.PF}{PA.PB.PC}$ Using areas relation, I found $\frac{3S_{ABC}-\overbrace{(S_{PAB}+S_{PAC}+...
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1answer
35 views

Solve for the area of a parallelogram; given diagonals and a side

Find the area of the parallelogram $ABCD$ with side $AB=10\sqrt{3}$ $cm$ and diagonals $BD=10\sqrt{3}$ $cm$ and $BC=10$ $cm.$ Using the fact that $AC^2+BD^2=2(AB^2+BC^2)$ we can find the other side ...
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4answers
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Find the radius of the circle given in the picture below.

This is the image of the question. I am not able to get how to find the radius. Please help with that. This is my try. I can't proceed now after it. Thanks
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Solving systems of equations by means of geometric construction

This is a follow up to this question: Constructing an equilateral triangle of a given side length inscribed in a given triangle My attempts to solve the problem were initially focusing on algebraic/...

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