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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.

2
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1answer
48 views

Prove that $\angle AED+\angle ADO=90^\circ$ given that $\angle BAC=60^\circ$.

Let ABC be a triangle such that $\widehat{BAC}=60^\circ$ and $AB\neq AC \neq BC$ and $O$ is the circumcenter of $ABC$. Let $D$ be the intersection of the internal angular bisector of $\widehat{BAC}$ ...
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0answers
14 views

How can we cut out $B'$ from $A$ such that $B'$ is similar to $B$ and $B'$ is as large as possible?

Given any arbitrary, closed shapes $A$ and $B$ on the Euclidian plane, how can we cut out $B'$ from $A$ such that $B'$ is similar to $B$ and $B'$ is as large as possible? This question, I found it in ...
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0answers
26 views

lower and upper bounds on $aA+ bB + cC$ and ${\large{\frac{ab}{l_c}}}+{\large{\frac{bc}{l_a}}}+ {\large{\frac{ca}{l_b}}}$.

Let $ABC$ be an acute triangle. The goal is to prove: \begin{align*} &(a)\;\;\;\pi(2R-r) < aA+ bB + cC < 4(2R-r)\\[2pt] &\qquad\qquad\text{[where in the above, $A,B,C$ denote the ...
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1answer
32 views

Equation of perpendicular line from the midpoint of a chord to a tangent on a unit circle (complex numbers)

As the title suggests, I would like the find the equation of the perpendicular line from $M_1$ of the side $A_2A_3$ to the tangent line at $A_1$. Without a loss of generality, suppose that on the unit ...
2
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2answers
57 views

The line $DN$ bisects the line segment $AC$ if $AD=BC$.

Consider a circle with diameter $AB$. A point $D$ on the circle is chosen arbitrarily such that $D\ne A,B$. A point $C\in AB$ is also chosen arbitrarily such that $C\ne A,B$. Draw $CH$ ...
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1answer
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Question on proving an isosceles triangles [on hold]

Let triangle ABC be isosceles with side AB congruent to AC. Suppose that points M and N are located on BC so that BMN*C (order of the points on side BC) and so that BM is congruent to CN. Prove that ...
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25 views

$P$ is a point inside the triangle $ABC$, from which perpendiculars intersect the sides $BC$ $AC$ $AB$ to the points…

$P$ is a point inside the triangle $ABC$, from which perpendiculars intersect the sides $BC$ $AC$ $AB$ to the points $D'$, $E'$, $Z'$. If $AD$, $BE$, $CZ$ are the heights prove that $$ \frac{PD'}{AD}+\...
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1answer
44 views

Can we solve a triangle given area and a perimeter? [on hold]

With two known data i can calculate the triangle's inradius. But then i have no idea what to do next to find three sides of the triangle.
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0answers
12 views

Prove intersection of PJ,NH lies in the nine-point circle

Given triangle $ABC$ and its altitudes $AD,BE,CF$ concur at $H$. Let $X,Y$ be the intersection of $(BE,FD), (CF,ED)$. Let $XY$ and the $D-$excircle of $\Delta DEF$ intersect the nine point circle at $...
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Why is euclidean geometry also called parabolic geometry?

Given that the three fundemental geometries are euclidean geometry (zero curvature), riemannian geometry (positive curvature) and lobachevskian geometry (negative curvature), I am curious as to what ...
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0answers
43 views

How to prove the Angle Addition Law for Sine using the Extended Sine Law? [on hold]

Extended Sine Law for $\triangle ABC$ with circumradius $r$: $$\frac{a}{\sin{A}}=\frac{b}{\sin{B}}=\frac{c}{\sin{C}} = 2 r$$ Angle Addition Law for Sine: $$\sin(a+b)=\sin a \cos b +\sin b\cos a$$ ...
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1answer
45 views

Kiselev's geometry Problem 67: In an isosceles triangle, two medians/bisectors/altitudes are congruent

Problem 67 from Kiselev's Geometry, Prove that in an isosceles triangle, two medians are congruent, two angle bisectors are congruent, two altitudes are congruent. Here is the content page in the ...
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0answers
60 views

How to find the center of an arithmetic spiral? Tangent to two lines

attempt at finding center of arithmetic spiral to be tangent to two lines Given the two orange lines in the picture, how can I calculate the center for an arithmetic spiral that will join the two ...
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0answers
55 views

Geometrical constructions of numbers

There are many ways to construct numbers (as points) with straightedge and/or compass systematically. Consider a reference curve $\mathcal{C}$ (a straight line or a circle), a set of points $\...
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0answers
12 views

Hessian Metric and Bregman divergence

I read from a paper that Bregman divergence is an approximation to the Hessian metric when the two points are nearby. What is the definition of Hessian metric? How can we derive this approximation?
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2answers
28 views

How to construct ray reflection from convex curved surface

I'm trying to understand how to construct the reflection path of a ray from a curved surface. Here's the basic setup: In a 2D space, assume a point S is the source of a ray and point R is the ...
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0answers
31 views

How were (any of the forms of) the equation of a line first derived?

How can I prove that the equation of a line, say in slope-intercept form, is $y=mx+c$? I'm trying to understand how the first person to come up with this equation did so.
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3answers
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Is the limit of this infinite step construction an equilateral triangle?

Just for fun (inspired by sub-problem described and answered here): Let's pick three points on a circle, say $A,B,C$. Move one point ($A$ for example) until the triangle becomes isosceles ($A'BC$) ...
2
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1answer
22 views

If angles $A$, $B$, $C$ of convex quadrilateral $\square ABCD$ are equal, then $D$ lies on the Euler line of $\triangle ABC$

In a convex quadrilateral $ABCD$ angles at $A,B,C$ are equal. Prove that vertex $D$ lies on the Euler line of triangle $ABC$. My try: We can use complex numbers. Set circumcirle of triangle $ABC$ as ...
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0answers
20 views

Barycentric coordinates of the centroid of parallelogram

Let ABCD be a non-collinear parallelogram, E be the midpoint of AB, and F be the midpoint of BC. Prove that D, E, F form an affine basis, and find the barycentric coordinates of the centroid of A, B, ...
2
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1answer
63 views

The line through certain points on extensions of two sides of a triangle is perpendicular to the line through the incenter and circumcenter

Extend side $AC$ of triangle $ABC$ to point $E$ such that $AB=AE$. Extend side $BC$ of triangle $ABC$ to point $D$ such that $AB=BD$. Denote the center of circumscribed circle of triangle $...
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0answers
62 views

Algebraic notation for geometric constructions

I wonder if there have been attempts to write the constructions that are performed in Euclidean geometry – drawing straight lines and circles defined by two given points, using specific ...
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2answers
63 views

Kiselev's geometry: Angles that have a common vertex

Quoted from Kiselev's geometry: Planimetry, page 18 section 27, (1) If the sum of several angles that have a common vertex is congruent to a straight angle, then the sum is 180 degrees. (2) If ...
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7answers
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Of all polygons inscribed in a given circle which one has the maximum sum of squares of side lengths?

My son presented me with an interesting problem: Of all possible polygons inscribed in a circle of radius $R$, find the one that has the sum $S$ of squared side lengths maximized: $S=a_1^2+a_2^...
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0answers
21 views

Proof of plane separation axiom on nonempty sets

I have the following question using the plane separation theorem: Let $l$ be a line and let $P$ be a point that is not on $l$. Consider two sets of points: $S_1 = P$ itself and all points on ...
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0answers
30 views

Need help in understanding how to solve for foot of perpendicular involving complex numbers

As the title suggests, I would like to solve for the foot of perpendicular involving complex numbers. In fact, I have already worked out the solutions, but I need help in understanding certain parts ...
1
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2answers
41 views

Angles in a nonagon

The following figure was draw by taking midpoints on three sides of a regular nonagon. The angles are 20 degrees, 50 degrees, and 50 degrees. It's not hard to find the angles by using trigonometry. ...
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0answers
34 views

Is there a general computational method to convert a Euclidean solid into the hyperbolic 3-space? [closed]

I have been working on modeling hyperbolic geometry and I was wondering if there was a methodology to convert some of my Euclidean solids, such as a cube, into the hyperbolic 3-space and vice versa. ...
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0answers
22 views

Euclidean Geometry: Parallel Postulate and transitivity of parallelism

Definition: Two lines are parallel if they are coplanar and everywhere equidistant. Postulate 2: Through a point in a plane not on a line, one and only one line can be drawn parallel to that line. ...
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2answers
23 views

Proving triangle congruence

I have been tasked to prove the following: $$\triangle ABC \cong\triangle EDC $$ Give that $C$ is midpoint of $\overline{BE}$, and angles $\angle B $ and $\angle E$ are right angles. How would you ...
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2answers
29 views

Is this a recommended approach to find the square of the sides of a triangle whose been divided by a median?

The problem is as follows: Figure 1 shows a triangle $\textrm{ABC}$ whose side $\textrm{BC = 4 > inches}$. It is known that $\textrm{AM}$ is a median whose length is equal to 1 inch. Find the ...
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3answers
156 views

How to find the sum of the sides of a polygon whose one vertex goes from the north of a circle and the other comes from the east in its perimeter?

The problem is as follows: In figure 1. there is a circle as shown. The radius is equal to 10 inches and its center is labeled with the letter O. If $\measuredangle PC=30^{\circ}$. $\textrm{Find AB+...
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0answers
23 views

Beautiful cyclic quadrilateral property involving isogonal conjugates.

Let $ABCD$ be a cyclic quadrialetral. Diagonals $AC$ and $BD$ intersect at point $S$. Denote midpoint of $AC$ with $M$. Choose points $P\in MD$ and $Q\in MB$ so that $PQ\parallel BD$ (in other words ...
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1answer
27 views

Questions related to the definition of the smallest affine plane

Here I have two questions related to affine planes. The smallest affine plane has four points and six lines where $$ \mathcal{P}=\{A, B, C, D\} $$ and $$ \mathcal{L}=\{\{AB\}, \{AC\}, \{AD\}, \{...
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Euclidean geometry as left invariant metric on Bianchi groups of type I or VII0

According to Wikipedia's article on the geometrization conjecture, one of the eight possible geometric structures for a manifold is the euclidean, or $E^3$. The article on Wikipedia says that this ...
3
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1answer
27 views

About Affine planes

I am studying about affine planes An affine plane can be defined as It is an ordered pair ($\mathcal{P}$ ,$\mathcal{L}$), P is non-empty set of points and and L is non-empty collection of the subsets ...
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2answers
39 views

Inscribed square in circle generates three collinear points.

Let $ABCD$ be a square inscribed in a circle $\mathcal{C}$ and P in $\mathcal{C}$ different from $A$. Assume that $PA$ intersects $BD$ at the point $E$. Let $M$ be the intersection of the line $PB$ ...
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1answer
29 views

How to find the distance to pixel ratio in a image.

I am trying to find a estimated distance in a image based on the camera height and angle. I know how to find the center of the image distance based off of this equation that I am using Camera Height ...
3
votes
1answer
26 views

Geometric intepretation of matrix vector product

Let $A = \begin{bmatrix} 1 & 0\\ 0 & 0\\ \end{bmatrix}$. The task is to find a constant $c$ and a vector $v$ (not zero vector) such that $Av = cv$ My attempt is to let $v =...
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1answer
41 views

How to algorithmically find only the edges of a high dimensional convex hull?

Given to me is a set of points $p_1,...,p_n\in\Bbb R^d$ in general position. I want to determine only the edges of the convex hull $C:=\mathrm{conv}\{p_1,...,p_n\}$, and this as efficient as possible. ...
3
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1answer
42 views

Compatibility of cross and inner product on $\mathbb{R}^3$

Consider the following operations on a three-dimensional vector space $\mathbb{R}^3$. The cross product $\times: \mathbb{R}^3 \times \mathbb{R}^3 \to \mathbb{R}^3$ can be defined by the following ...
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1answer
27 views

A convex quadrilateral with propotion

Let P be the intersection of the convex quadrilateral ABCD. Let X,Y,Z be points on AB,BC,CD respectively such that $\frac{AX}{XB}=\frac{BY}{YC}=\frac{CZ}{ZD}=2$. Suppose that XY is tangent to the ...
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2answers
50 views

Another beauty hidden in a simple triangle (3)

In an arbitrary triangle $ABC$ pick arbitrary points $D\in BC$ and $E\in AC$ such that $DE \nparallel AB$. Denote midpoint of segment $BD$ with $F$ and midpoint of segment $AE$ with $G$. Now draw ...
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5answers
1k views

Covering the Euclidean plane with constructible lines and circles

It is a well-known fact that the set of points which are finitely constructible with straightedge and compass (starting with two points $0$ and $1$) doesn't cover the Euclidean plane $\mathbb{R}\times ...
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1answer
30 views

$O$ is the orthocentre of $\triangle ABC$ if $AP\perp BC$, $BR\perp AC$ and $CQ \perp AB$. Prove that $\angle OPQ= \angle OPR$ [closed]

$O$ is the orthocentre of $\triangle ABC$ if $AP\perp BC$, $BR\perp AC$ and $CQ \perp AB$. Prove that $\angle OPQ= \angle OPR$
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1answer
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Let $PAB'C, PBC'A, PCA'B$ form parallelograms so there exists a point $Q$.

Let $P, A, B, C, A', B' , C' \in \mathbb{R}^2$ be points such that $PAB'C, PBC'A, PCA'B$ form parallelograms. Prove that there is a unique point $Q$ so that $QA'BC' , QB'CA' , QC'AB'$ form ...
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0answers
16 views

In multiobjective optimization, how to calculate the distance to reference point?

In multiobjective optimization, what does the distance exactly means, is it: 1) The distance from reference point (V) to an individual (Xi) (candidate solution) in the population (decision space). <...
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2answers
39 views

Basic similarity of triangles problem

The question is to find the length of AB. I'm currently studying civil engineering, but I'm trying to refresh some old knowledge as I got a part time job this semester as a math assistant at a middle ...
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2answers
100 views

Prove that $D$ is the incentre of the triangle $IJK$.

Let $ABC$ be a triangle and $S$ its circumcircle. The points $D$, $E$, and $F$ are the feet of altitudes drawn from $A$, $B$, and $C$, respectively. The line $AD$ meets $S$ again at $K$. Let $M$ and ...
1
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1answer
37 views

Prove $|AD|^2=|AB|\cdot |AC|-|DB|\cdot |DC|$ in triangle $ABC$, D the point where bisector of angle $A$ intersects $\overline {BC}$

In a triangle $\triangle ABC$, the bisector of angle from the point $A$ intersects $\overline {BC}$ in point $D$. Prove: $|AD|^2=|AB|\cdot |AC|-|DB|\cdot |DC|$. I don't even know where to start. I'll ...