geometry assuming the parallel postulate of Euclid: 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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1answer
23 views

Chord and tangent properties [on hold]

In a cyclic quadrilateral ABCD, the diagonal AC bisects the angle BCD. Prove that diagonal BD is parallel to the tangent to the point circle at point A.
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13 views

An integral from the integral geometry about the isoperimetric inequality.

The problem is from the book "Integral Geometry and Geometric Probability" by Santalo (1976), Chapter 1.3.5, Notes and Exercises (page 37). Given a convex closed curve $C$. Let $A_1$, $A_2$ be the ...
5
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1answer
95 views
+50

Space Geometry: lines in a plane

If $d$ and $d'$ are two intersecting lines in a plane $P$, and $D$ is a line orthogonal to both $d$ and $d'$, then any line $\delta$ in $P$ is orthogonal to $D$ as well. How could this be proven ...
26
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4answers
964 views

Two squares in a box.

According to Arthur Engel, "Problem Solving Strategies", this problem goes back to Erdős, but I cannot find the solution: Let $A$ and $B$ be two non-overlapping squares inside a unit square, of side ...
0
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1answer
39 views

Prove base times altitude is a constant without resorting to area.

The area of a triangle is one-half base times altitude. This implies that, for $\bigtriangleup ABC$, $ah_A = bh_B=ch_C$, where $h_A$ is the length of the altitude dropped from point A to side BC, etc. ...
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1answer
19 views

What's the name given to the ratio $P^2/A$ for a closed figure in the Euclidean plane?

Let $\mathscr{F}$ be the set of all plane, closed Euclidean figures having positive perimeter, and let $\sim$ be the similarity relation on $\mathscr{F}$. Then, for any equivalence class ...
10
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1answer
1k views

Construction of a regular pentagon

In Robert Dixon's Mathographics, a regular pentagon is constructed with straightedge and compass only. It is the pentagon $ABCDE$ pictured below. I am having trouble seeing why the central angles ...
5
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2answers
375 views

Inscribing a quadrilateral inside a rhombus

Let ABCD be a rhombus, its interior angles are $\alpha<\frac{\pi}{2}$ and $(\pi-\alpha)$. Let w, x, y, z four points located respectively in (A, B), (B, C), (C, D), (D, A). Suppose we have as ...
2
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2answers
359 views

Algebraic solution to find circle radius given distance of three external points from perimeter

I have an engineering problem, which involves math. The reason it's "engineering" is that I don't need a pure mathematical solution, but a good-enough approximation could work - the only constraint is ...
0
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0answers
24 views

About Homothetic transformation

I have one question regarding the way which a homothetic transformatior is written. Why is it written in the following way: $$\vec{OH^{k}_{O}(P)}=k\vec{OP}+(1-k)\vec{OO}?$$ From where ...
5
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4answers
2k views

At what coordinates should the fourth vertex be located?

A parallelogram is drawn on a coordinate grid so that three vertices are located at $(3, 4)$, $(-2, 4)$ and $(-4, 1)$. At what coordinates should the fourth vertex be located?
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1answer
645 views

find the center of an ellipse given tangent point and angle

I have an ellipse with known major radius $r_x$ and minor radius $r_y$, aligned with the x- and y-axis. Given a tangent point $T$ and the tangent angle $\alpha$, how do I calculate the center $C$ ...
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3answers
62 views

Find the area of triangle, given an angle and the length of the segments cut by the projection of the incenter on the opposite side.

In a triangle $ABC$, one of the angles (say $\widehat{C}$) equals $60^\circ$. Given that the incircle touches the opposite side ($AB$) in a point that splits it in two segments having length $a$ ...
3
votes
1answer
96 views

What do the Purser's theorem says?

Mathworld's entry on Purser's Theorem says the following: Let $t, u$, and $v$ be the lengths of the tangents to a circle $C$ from the vertices of a triangle with sides of lengths $a, b$, and $c$. ...
0
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1answer
119 views

The Nearest Points

Given a set $R$ of $N$ points $R={(x_1, y_1, z_1), (x_2, y_2, z_2),....., (x_n, y_n, z_n)}$ and set $S$ of $M$ points $S={ ((a_1, b_1, c_1), (a_2, b_2, c_2),...(a_m, b_m, c_m))}$. for each point ...
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1answer
31 views

Equivalent definitions of symmetry group of regular n-gon (dihedral group)

Let $P_n$ be a fixed regular convex $n$-gon in the plane. For a metric space $M$ we denote by $\text{Isom}(M)$ the set of distance-preserving maps $M \to M$. How can I show that $$ D_n := \left\{\, f ...
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2answers
25 views

Does being distance-preserving force affinity?

Let $\phi: \mathcal{E}\to \mathcal{E}$ be a mapping defined on a Euclidean affine space. Assume that $\phi$ preserves the distances: $$d(\phi(A),\phi(B)) = d(A,B).$$ Prove that $\phi$ is affine, and ...
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3answers
3k views

Geometric Definitions: What is a straight line? What is a circle?

What is a straight line? I need a geometric definition of it. The equation of a straight line is known to me.I am saying about a straight line of 2D plane. What is a circle? I need a geometric ...
4
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0answers
187 views

Which chapters of Euclid's elements would be helpful for drawing a grid?

I am drawing a $19 \times 19$ grid on my desk. For aesthetic purposes, I don't want to use a ruler. Rather, I want to use Euclidean theorems to 'prove' to myself that such and such line meets at a ...
0
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1answer
25 views

Why is not parity transformation just a rotation?

I'm a bit confused about parity transformations (reflections). A parity operator $\pi$ takes a vector $(x, y, z)$ to $(-x, -y, -z)$. So in a $3$ dimensional space this takes a vector and points it ...
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1answer
50 views

Proving Ordering of Angles

I'm trying to prove $$\text {If}\ \angle P \lt \angle Q, \text & \ \angle Q \cong \angle R, \text{Then}\ \angle P \lt \angle R$$Which seems super basic and makes sense, but I got told that I'm ...
16
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1answer
262 views

With what analytic functions can one construct the $(x,y)$ coordinate axes using a straightedge and a compass?

Given the graph of $y = \frac{1}{x}$, construct the $(x,y)$ coordinate axes using a straight edge and a compass. The solution to this problem is known (mouse over the spoiler text below for a ...
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2answers
44 views

Question about the existence of points and lines.

Say we draw a point on a graph. If the point should not take up any area than how come we could see it. Say we graph $y=x^2$, we obviously could see it. However, because $y=x^2$ is a function made up ...
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2answers
28 views

Find 3D distance between two parallel lines in simple way

Is there a simple way to get 3D distance between two parallel lines given end points of each line?
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2answers
246 views

Problem in understanding a proof there are five Platonic solids.

Thanks to several comments by Gerry Myerson, it is now clear that I wasn't clear, up to a state where I seriously confused myself. In a renewed attempt: Recently, I've been thinking about Platonic ...
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2answers
60 views

All right angles are equal to each other

Why is it that All right angles are equal to each other -a postulate in Euclid's Elements (Wikipedia). Shouldn't it be a congruence rather than an equivalence? Isn't this just a special case of ...
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1answer
22 views

The projective space of all lines through the origin

I have a question to the following example: Assume that $\mathbb{A}_2$ is an affine plane over a field $\mathbb{K}$, and we have fixed affine coordinates $x, y$ on $\mathbb{A}_2$. Let $\mathbb{P}$ be ...
16
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4answers
1k views

The position of a ladder leaning against a wall and touching a box under it

I was reading a newspaper and there was a little math riddle, I thought "how funny, that's gonna be easy, let's do it" and here am I... The problem goes as follow : in a barn, there is a 1 meter ...
3
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5answers
52 views

Showing that certain points lie on an ellipse

I have the equation $$r(\phi) = \frac{es}{1-e \cos{\phi}}$$ with $e,s>0$, $e<1$ and want to show that the points $$ \begin{pmatrix}x(\phi)\\y(\phi)\end{pmatrix} = ...
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0answers
27 views

Pascal's theorem in ellipse

I know that the proof of Pascal's theorem in circle is based on the power of a point. I also heard that one can use projective geometry to prove it in an ellipse. But is there a direct synthetic proof ...
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1answer
84 views

Find the Height of the Trapezoid

Problem: The area of a trapezoid is equal to 2 and the sum of his diagonals is equal to 4. Find the trapezoid height. [QUESTION]: I find a result that implies that the height of the ...
3
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2answers
63 views

Construction of a circle through a point and tangent to angle

given an angle $\angle (h,k)$, where $h,k$ are the legs of the angle. Let $P$ be some point in the interior of the angle. I want to construct a circle through P which is tangent to both legs $h,k$. ...
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2answers
25 views

Incommensurable line segments

I have an issue with a proof given in my lecture hopefully someone can help me with. It will be shown that the hypotenuse $c$ of a right-angled and isosceles triangle will be incommensurable to the ...
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1answer
41 views

Two prove two lines in a triangle are parallel

$D$, $E$, $F$ are the midpoints of sides $BC$, $CA$ and $AB$ respectively of a triangle $ABC$ right angled at $C$. If $EF$ and $DE$ (extended if necessary), meet the perpendicular from $C$ on $AB$ in ...
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3answers
46 views

Is it possible to find the vertices of an equilateral triangle given its center point?

I was wondering how to find the vertices of an equilateral triangle given its center point? Such as: ...
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0answers
26 views

tessellation of an arbitrary shape [closed]

Are there any shapes that we can tessellate any plane shapes (with arbitrary shapes) by them? i.e. if I generate a random shape, how can I tessellate it by some shapes?
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1answer
19 views

Points on two skew lines closest to one another

Given two skew lines defined by 2 points lying on them as $(\vec{x}_1,\vec{x}_2)$ and $(\vec{x}_3,\vec{x}_4)$. What are the vectors for the two points on the corrwsponding lines, distance between ...
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0answers
14 views

Dropping parallel postulate and infinitude of straight line

I was reading an article about back ground of Killing's work by Thomas Hawkins from Historia mathematica 1980.It was written that Killing stated that if one drop assumptions infinitude of straight ...
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1answer
30 views

The limit incentral triangle is equilateral [closed]

I found a nice problem of geometry but I don't know how to prove it. Given a triangle $T_0$, we build $T_1$ by considering the projections of the incenter of $T_0$ on the sides of $T_0$. In the ...
0
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1answer
36 views

Construct the triangle with given angle bisectors

given three lines $\ell_1,\ell_2, \ell_3 $ which intersect in one point $P$. How can one construct a triangle such that the given lines become its angle bisectors? So far I tried to find conditions ...
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0answers
20 views

To circumscribe a square about a given circle.

http://aleph0.clarku.edu/~djoyce/java/elements/bookIV/propIV7.html I was just wondering something . We know that if a line touches a circle at one point, then this means that this line is forming a ...
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1answer
305 views

A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
13
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1answer
287 views

How big is my pizza, if I know its slices' sizes?

I bought a box of frozen pizza: eight slices, baked and then frozen, stacked in a box. The packaging assured me that I was holding an 18-inch[-diameter] pizza. That got me thinking: how do I know ...
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1answer
33 views

Construction of triangle from side $c$ and heights $h_a, h_b$

I want to construct a triangle $\Delta(A,B,C)$ with given side $c$ and heights $h_a, h_b$. To construct the triangle means to use only ruler and compass. How can I solve this? I started as follows: ...
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1answer
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I am looking for a function in order to measure points misalignment

The points are in the euclidean plane, let $\mathbb{P}$ be the set containing all the finite sets of $\mathbb{R}^2$ points. I am looking for a function $m : \mathbb{P} \to \mathbb{R}^+$ in order to ...
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1answer
355 views

Internal polygon formed by drawing diagonals in a regular polygon

In an n-sided (n>4) regular polygon, label the vertices {0, 1, ..., n-1}. For each vertex i, draw a pair of diagonals: from i to (i+2) mod n and from i to (i-2) ...
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14 views

Planar nearest neighbor search for many points.

I have two sets of points on the plane, A and B. For every point in A, I would like the k nearest points in B. The naive approach is for each point in A having a linear selection to choose the kth ...
3
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2answers
82 views

Let $ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$. Determine $S+S+…+S $.

Let $$ S=\{(x,y)\in\mathbb{R}^2 \ | \ x^2+y^2=1 \text{ and } y\geq 0\}$$ By the usual notation for sum of sets let $$ 2S\overset{\text{not}}{=}S+S=\{(x_1+x_2,y_1+y_2) \ | \ (x_1,y_1), ...
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22 views

Name of the segment connecting a point's coordinate axis projections?

Given any point $(x,y)$ in the real plane consider the corresponding line segment connecting $(x,0)$ with $(0,y)$. See diagram. Is there a name for this special segment? (I believe that in ...
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8 views

Identifying Peak Points in a 3-dimensional space

I have a data set which is composed of $xyz$ points. I want to be able to identify peak and valley points from the set. Identifying peak for 2D points $(x,y)$ is simple. I want to know what properties ...