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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4
votes
4answers
265 views

A problem concerning a parallelogram and a circle

Sorry for the ambiguous title. If you can phrase it better, feel free to edit. "A parallelogram $ABCD$ has sides $AB = 16$ and $AD = 20$. A circle, which passes through the point $C$, touches the ...
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1answer
21 views

Affine space definition

I am studying affine space But when i see its definition on YouTube it looks like Let $A$ is set of points and $V$ is vector space over R and $f:A×A\to V$ such that $\forall$ $x, y, z\in A $ $f(x,y)+f(...
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1answer
21 views

Area of a rectangle using congruency

I don't understand the lecturer, solving the question, says that the rectangles are congruent each other, so the result can be obtained by proportioning them to each other. However, AFAIK two ...
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2answers
38 views

Crux problem #39 with vectors approach

It's given a point $P$ inside an equilateral triangle $ABC$ such as segment lengths $PA$, $PB$, $PC$ are $3$, $4$ and $5$ units respectively. Calculate the area of $\triangle ABC$. So, let $b:=\...
2
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2answers
105 views

Geometry question: Find the area of blue-shared area inside this isosceles

See below, looks a bit interesting, but I cannot find a solution. I think a starting point might be the similarity of the lower white triangle and the larger triangle composed of lower blue, pink, and ...
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1answer
21 views

Show one diagonal of $B D E C$ divides the other diagonal in the ratio?

Consider a triangle $A B C$. The sides $A B$ and $A C$ are extended to points $D$ and $E,$ respectively, such that $A D=3 A B$ and $A B=3 A C$. Then one diagonal of $B D E C$ divides the other ...
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0answers
55 views

$3$ points are collinear

$M$ is a point on $AB$. $BMC$ and $AMD$ are constructed such that they are both equilateral & on the same side of $AB$. The circumcircle of both triangle intersect at $M$ and $N$. Prove that $AD$ ...
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1answer
41 views

Suppose $AD$ is a height of $\triangle ABC$ and $H$ is the orthocenter. Is it true that $BD \cdot DC = AD \cdot DH$?

Suppose $AD$ is a height of $\triangle ABC$ and $H$ is the orthocenter. Is it true that $BD \cdot DC = AD \cdot DH$? I sense that it is false and I have tried to find a counterexample, but I cannot ...
0
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1answer
35 views

Property of cyclic quadrilaterals [closed]

Suppose $ABCD$ is a cyclic quadrilateral and $P$ is the intersection of the lines determined by $AB$ and $CD$. Show that $PA·PB= PD·PC$ Could you help me please, I have no idea how to relate the ...
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0answers
41 views

Prove BMXN is cyclic.

Suppose $C_1$ and $C_2$ are circles such that {$𝐴,𝐵$}=$𝐶_{1}\cap 𝐶_2$. We draw a secant $MN$ such that $𝑀\in 𝐶_1$ and $𝑁\in 𝐶_2$, and $A\in MN$. Show that if $X$ is the point of intersection ...
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1answer
90 views

Given the feet of the altitudes of $\triangle ABC,$ point $R$ and the midpoint $P$ of $\overline{AB}$, prove$ |RA|\cdot|RB|=|RP|\cdot|RN|$

Let $K,M,N$ be feet of the altitudes of $\triangle ABC$ from the vertices $A,B,C$ respectively, $P$ be the midpoint of the edge $\overline{AB}$ and $R$ be the intersection point of the lines $AB$ and $...
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3answers
76 views

Constructing a right triangle with a given hypotenuse segment and given point of tangency for its incircle

Given a hypotenuse $AB$ and an arbitrary point $C$ on $AB$. How to construct a right triangle with the given hypotenuse $AB$ such that point $C$ is the point of tangency of the inscribed circle? My ...
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0answers
19 views

Simple tilted perspective projection

In 3D Space denoted by coordinates x,y,z, I have $2D$ coordinates X,Y on a plane that is constructed from the x,y-plane (z=0) by rotating it around the $x-$axis by $\phi$ degrees. I want to do ...
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4answers
54 views

Sum of distances of a variable point from two fixed points.

Given that $A\equiv(4,2)$ and $B\equiv(2,4)$, find a point $P$ on the line $3x+2y+10=0$ such that $PA+PB$ is minimum. My attempt: In $\triangle PAB$, $$PA+PB\ge AB$$ Hence, the minimum value should ...
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1answer
31 views

Line $CD$ meets the leg $\overline{AB}$ at $P$ and line $BF$ meets the leg $\overline{AC}$ at $Q$. Prove $|AP|=|AQ|$.

$\triangle ABC$ is a right-angled triangle at $A$. There are squares $AEDB$ and $ ACFG$ described from the outside on the legs $\overline{AB}$ and $\overline{AC}$, respectively. Line $CD$ meets the ...
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0answers
30 views

Reflection of light from a vertex of of a cube to another

Let $ABCD$ and $BCFG$ be two faces of a cube with $AB=12.$ A beam of light emanates from vertex $A$ and reflects off face $BCFG$ at point $P,$ which is 7 units from $\overline{BG}$ and 5 units from $\...
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7answers
274 views

Crux problem #33 with vector approach

On the sides $CA$ and $CB$ of an isosceles right-angled triangle $ABC$, points $D$ and $E$ are chosen such that $|CD|=|CE|$. The perpendiculars from $D$ and $C$ on $AE$ intersect the hypotenuse $AB$ ...
1
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1answer
41 views

One altitude, one bisector and one median are equal implies equilateral triangle

Let $ABC$ be a triangle such that the lenghts of the $A$-angle-bisector, the $B$-median and the $C$-altitude are equal. Prove that $ABC$ is equilateral. I was only able to show that $AB$ is the ...
2
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1answer
40 views

Area of Projection to a Plane

I need to find all planes of the form $ax+by+cz=0$ such that the projection of: $$ S=\left\{x^2+y^2+z^2=4\middle|x^2+y^2\leq 2x\right\} $$ Onto that plane is one-to-one, and then use it somehow to ...
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1answer
38 views

Expected maximum sum of pairwise distance for *n* points on a cylinder

I'm trying to calculate a measure of 'concentration' or 'clusteredness' for n points on a cylinder. Say I start with a plane of width=12 (x) and height=10 (y) on which 10 points are situated. These ...
2
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1answer
36 views

Proving inequality of line segments in a circle

There are 2 intersecting circles with the centers $O_1$ and $O_2$ and the radiuses $r_1$ and $r_2$ respectively ($r_1 \gt r_2$). They have a common segment line $AB$. Also $AC$ is the tangent line of ...
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0answers
44 views

Maximum number of acute triangles in a polygon convex

If $𝑃$ is a convex polygon and is divided into $𝑛 − 2$ triangles with diagonals. What is the maximum number of acute triangles you can have? I did not understand the part of triangles with diagonals,...
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1answer
31 views

Mapping two superquadratics

Superquadrics are a family of geometric shapes defined by $$|\frac{x}{A}|^r + |\frac{y}{B}|^s + |\frac{z}{C}|^t =1$$ I have two superquadratics, namely SQ1 and SQ2, with parameters $\{A_1,B_1,C_1,r_1,...
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1answer
61 views

A question on Triangle Inequality in $\mathbb{R}^n$

I'm reading a textbook on Topology. We know that $(\rho,\mathbb{R}^n)$ is a metric space, where $$\rho(x,y)=\sqrt{\sum_{i=1}^n(x_i-y_i)^2}$$for any $x=(x_1,x_2,\ldots,x_n),y=(y_1,y_2,\ldots,y_n)\in\...
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3answers
64 views

In $\Delta ABC$, if $2\cos\frac{A-C}{2}=\frac{a+c}{\sqrt{a^2+c^2-ac}}$, then choose the right options

$B=\frac{\pi}{3}$ $B=C$ $A, B, C$ are in arithmetic progression $B+C=A$ $$2\cos \frac {A-C}{2}=\frac{\sin A+\sin C}{\sqrt{\sin^2 A+\sin^2 C -\sin A\sin C}}$$ $$\sqrt {\sin^2A+\sin^2C-\sin A\sin C}=\...
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5answers
113 views

Maximum number of acute triangles in a convex polygon

I have been reading about convex polygons and I found the following: We say that a simple polygon is convex if all its interior angles are less than $\pi$. If $P$ is a convex polygon and is divided ...
0
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0answers
32 views

$\exp(x)-\exp(y)$ coordinate system embedded in $\Bbb R^2$

Define a coordinate system in $\Bbb R^2$ with an exponential scale on the axes: $\exp(x)-\exp(y).$ Then embed this coordinate system in the first quadrant of $\Bbb R^2$ such that we view the axes now ...
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0answers
99 views

How does spherical geometry contradict Euclid's parallel postulate?

Euclid's parallel postulate says: 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 ...
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0answers
46 views

Prove$\frac{1}{PA_1^2}+\frac{1}{PA_2^2}+\cdots+\frac{1}{PA_n^2}=\frac{n^2}{4R^2}.$

Let $A_1A_2\cdots A_{n}$ be a $n$ regular polygon with a circumscribed circle of radius $R$, and $P$ be the midpoint of the inferior arc ${A_1A_n}$. Prove that $$\frac{1}{PA_1^2}+\frac{1}{PA_2^2}+\...
-2
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1answer
48 views

RMO 1990 question [closed]

Prove that the inradius of a right angled triangle having integer sides is also integral I tried it and got something like $r=\frac {(a.b)}{(a+b+c)}$ How to proceed after this.
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2answers
48 views

In which ratio does the point $P$ divide the segment $\overline{AN}$?

In an arbitrary triangle $\triangle ABC$, let $M\in\overline{AC}$ s. t. $|AM|:|MC|=2:1$ and let $N\in\overline{BC}$ s. t. $|BN|:|NC|=1:2$. Let $P$ be the intersection point of the segments $\overline{...
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1answer
57 views

Finding angles in quadrilateral.

Let $ABCD$ a quadrilateral, $AC\cap BD=\{O\}$, $\angle A=110^{\circ}$, $\angle DOC=60^{\circ}$ and $AC=AB+CD=BD$. Find $\angle B$, $\angle C$. I tried a lot of constructions. I think $ABCD$ is a ...
2
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1answer
47 views

Algorithm to compute “median”

Let $x_1,\dots,x_m\in\mathbb{R}^d$ be a finite set of points. I define the $d$-dimensional "median" $y\in\mathbb{R}^d$ to be the point minimizing the sum of distances to $x_j$, $$ y = \arg\...
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1answer
29 views

Square ABCD of side a and N on AB. Find radius of congruent incircles of ACN and BCN [closed]

ABCD is a square with side a and diagonal AC. The incircles, with radius r, of the triangles ACN and BCN are congruent, with N in AB. What is the radius r in terms of a?
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1answer
23 views

Equal distance products [closed]

Prove that in every inscribable quadrilateral, the product of the distances from any point on the circumscribed circle circumference to two opposite sides is equal to the product of the distances from ...
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2answers
178 views

Complex Euclidean Geometry Question

Let w be the incircle of a fixed equilateral triangle ABC. Let l be a variable line that is tangent to w and meets the interior of segments AC and BC at P and Q, respectively. A point R is chosen such ...
6
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1answer
49 views

Distance from one vertex of a rectangular prism to the plane determined by three other vertices, without using vectors or calculus

The objective is to find the shortest distance from the point $H$ to the plane $BDE$. The prism $ABCD.EFGH$ has $AB=AD=5\sqrt{2}$ and $AE=12$. I think that these numbers are badly selected by the ...
0
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1answer
38 views

In $\triangle ABC$, $D$ bisects $AB$, $E$ trisects $BC$ and $\angle ADC=\angle BAE$. Find $\angle BAC$

Giving a $\triangle ABC$ having point $D$ as the mid point of $AB$ and $E$ as the trisection point on $BC$ such that $BE>CE$. If $\angle ADC=\angle BAE$. Find $\angle BAC$ I can't solve this ...
4
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1answer
102 views

how to prove the segment $FH=HE$

Given a right-triangle $\triangle AGC$ $(\angle AGC=90^\circ)$ point $D$ is an arbitrary point on the altitude. $AE=AG$ and $CF=CG$. How to prove that $FH=HE$? I found that circles centered at $A$ ...
1
vote
1answer
49 views

Counting the number of lines

Show that there exists a constant $A$ such that for any $n$ points in $\mathbf R^2$, when $2\le k\le \sqrt n$, there are at most $\frac{An^2}{k^3}$ lines passing through at least $k$ points. ...
2
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0answers
74 views

How to inscribe a square in an arbitrary quadrilateral using compass and straight edge

Is it possible to inscribe a square in an arbitrary convex quadrilateral $ABCD$ with only compass and straight edge? I know how to construct a square inscribed in a triangle, but I don't know how to ...
0
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3answers
52 views

Intersecting triangles: Find the length of a segment given one side and two ratios

I'm trying to find the length of the segment from $H$ to $J$. The problem states the length of the segment from $B$ to $E$ is $494$, along with a couple ratios of the sides. $BC:CD = 2:3$ and $DE:EF:...
0
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1answer
38 views

Finiding orientation of a rod

I have rod made with small spheres like this. All the spheres together make this rigid rod.I have coordinate of all small spheres in 3d thus I can be able to find out it's center of mass and most ...
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1answer
56 views

Elementary Geometry : Structure of presentation

I will have a presentation on elementary geometry, and more precisely on the straight line and the triangle. The presentation should be in the university, in front of the fellow students. For this I ...
2
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1answer
37 views

Find all the parameters $\lambda\in[0,1]$ s. t. $\measuredangle BTQ=90^\circ$.

Let $\triangle ABC$ be equilateral with the side length $1$, $P$ be the midpoint of $\overline{AB}$ and $Q\in\overline{AC}$ s. t. $\overrightarrow{AQ}=\frac13\overrightarrow{AC}$. Let $T$ be a point ...
0
votes
3answers
40 views

Find the minimum perimeter of the triangle.

Consider point $A(5, 2)$ and variable points $B(a, a)$ and $C(b,0),\, a\in R, \, b\in R$. If the perimeter of $\triangle ABC$ is minimum, find $a$ and $b$. My attempt: $\begin{align} P&=AB+BC+CA\\...
1
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1answer
49 views

Calculate, in cm, the length of segment KL

Let ABCD be a rectangle such that AB = $ \sqrt {2} BC $. Let E be a point on the semicircle with diameter AB, as shown in the following figure. Let K and L be the intersections of AB with ED and EC, ...
4
votes
1answer
76 views

Geometry problem I am having trouble to solve

Prove that $AC = \sqrt{ab}$ $a$ is $AB$; $b$ is $CD$; the dot is the origin of the circle. ABCD is a trapezoid, meaning AB || DC. My attempt at solving: According to this rule, $$MA^2 = MB \cdot MC$...
1
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0answers
56 views

Is there any proof that Euclid's fifth postulate cannot be proven? [duplicate]

I've read many times that a plethora of people have tried to prove Euclid's parallel postulate but to no avail. However, I do not know of any proof of how it cannot be proven. Is it because modifying ...
0
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1answer
34 views

Combination of reflection symmetries in $\mathbb{E^4}$

Is the combination between point reflection (https://en.wikipedia.org/wiki/Point_reflection) symmetry and hyperplane(or axial using Hodge duality) reflection symmetry (https://en.wikipedia.org/wiki/...