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
77 views

Finding the angles of $\triangle ABC$

In a $\triangle ABC$, from vertex $C$, the median to $AB$, the angle bisector of $\angle BCA$ and the perpendicular to $AB$ divides angle $\angle BCA$ into four equal parts. The task is to compute ...
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2answers
12 views

Compute ratios $\frac{XP}{PY}$ and $\frac{BQ}{QC}$ in terms of lengths r=$\overline{AX}$, s=$\overline{XB}$, t=$\overline{AY}$ and u=$\overline{YC}$

Show that the only way these ratios can be equal is if both ratios equal one, and show that in that case, $\overline{XY}$ is parallel to $\overline{BC}$. I have ...
1
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1answer
21 views

The three Cevian's are concurrent at point T. Show that all six small triangles have equal areas.

It is given that $\Delta ART$, $\Delta BPT$, and $\Delta CQT$ have an area of one. I have no idea how to approach showing that all of the triangles have an equal area of one. I shaded the three ...
2
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0answers
23 views

Equation curves

Introduce equation curves to the canonical form, finding an appropriate rectangular coordinate system. a) $5x^2+12xy-22x-12y-19=0$ b) $9x^2+24xy+16y^2-230x+110y-475=0$ Could somebody do one task. I ...
2
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1answer
24 views

Misunderstanding in the following proof about ellipses.

While studying ellipses, I've read the following proof which I can't understand. Taking two point $F_1$ and $F_2$,we can define an ellipse as the set of points $Z$ sucht that $ZF_1+ZF_2$ has ...
4
votes
2answers
71 views

Rotation of a regular tetrahedron

The tetrahedron can be written with its apex at the north pole of a sphere with the four vertices: \begin{eqnarray} a(0,0,\sqrt{6}/4) \; , \; a(\sqrt{3}/3, 0, -\sqrt{6}/12) \; , \; a(-\sqrt{3}/6, ...
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2answers
44 views

How to prove this theorem rhetorically?

It is not possible for a part of any of three conic sections to be an arc of a circle. It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever ...
4
votes
1answer
44 views

Hexagon tesselations

A configuration is made of congruent regular hexagons,where each hexagon shares a side with another hexagon. What is the largest integer $k$, such that the figure cannot have $k$ vertices ? For ...
0
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1answer
23 views

Determine coordinate of a point on unit sphere

Let $S$ be unit sphere in $\mathbb R^3$ center at $O(0,0,0)$. Let $A=(x_1,y_1,z_1),B = (x_2,y_2,z_2)$ be two points lying on the sphere $S$. Let $M$ be center of $AB$ which lies on the geodesics ...
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0answers
17 views

Find the largest rectangle aligned with a side of a quadrangle

My problem finds a concrete application if you want to install a rectangular window EFGH in a near-rectangular hole ABCD (for example you want to build a metallic frame for an existing building where ...
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votes
5answers
311 views

Prove that $\angle FGH = \angle GDJ$

Let $FGH$ be a triangle with circumcircle $A$ and incircle $B$, the latter with touchpoint $J$ in side $GH$. Let $C$ be a circle tangent to sides $FG$ and $FH$ and to $A$, and let $D$ be the point ...
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0answers
44 views

Orthogonality of vectors and its dependence on the inner product.

Consider a set of vectors, $\{{\bf e}_i\}$ in $\mathbb{R}^n$. I am thinking specifically of the standard orthonormal basis. I am having a very difficult time understanding what it means for vectors to ...
0
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0answers
22 views

Using Affine Transformation to prove Concurrency

Let $ABCDE$ be a convex pentagon with $F=BC\cap DE, G=CD\cap EA, H=DE\cap AB, I=EA\cap BC, J=AB\cap CD$, Suppose that the areas of $\triangle AHI, \triangle BIJ, \triangle CJF, \triangle DFG, ...
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1answer
14 views

Similiar triangle inside a triangle based off of segment from feet of altitudes.

Given some triangle ABC, with feet of altitudes D, E, and F, I need to show that triangle ABC is similiar to triangle AEF. This is an image I made for it I have been able to show that the angles of ...
0
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1answer
48 views

sum of distances in acute-angled triangle

To each point in the sides and the interior of an acute-angle triangle, what is the maximum sum of the distance of these points to the 3 sides of the triangle? a) the arithmetic mean of the 3 heights ...
0
votes
1answer
18 views

minimization of sum of distances using elementary methods

I want to find a solution to the following minimization problem using only elementary methods. This is to say: high school algebra, basic inequalities, basic trigonometry and trigonometrical ...
1
vote
1answer
55 views

Rotation Arc Length in 4D

If I have a point $(x_0,y_0,z_0,w_0) \in \mathbb{R}^4$ and I rotate it within the $xy$-plane ($0<\alpha<\pi$) and $zw$-plane ($0<\beta<\pi$), how can I determine the length of the arc ...
0
votes
1answer
39 views

Distance between points at circles

May be someone can help me to solve the problem. There are circle with radius R1 and circle with radius R2. We also know the distance between A and O and that angle AOB = $\phi$. The aim is to ...
2
votes
2answers
51 views

For every three points on a line, does there exist a triangle such that the three points are the orthocenter, circumcenter and centroid?

The Euler line states that the orthocenter, circumcenter and centroid of a given triangle are on one line. This made me wondering whether the following is true: For every three points on a line ...
0
votes
2answers
23 views

Given $u,v\in \mathbb{S}^n$, then there exists a orthogonal matrix s.t. $u=O v$.

Is the following intuitive statement true: For given $u,v \in \mathbb{S}^n$, there exists a matrix $R\in O(n)$ s.t. $u=Rv$.
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1answer
17 views

Prove/disprove statement about functions with special properties

Suppose $H:\mathbb{R}^{2n}\to\mathbb{R}$ satisfies the following: $H\geq0$ everywhere and $H(x)=0\iff x=0$; $H\in\mathcal{C}^2(\mathbb{R}^{2n}\smallsetminus\{0\})$; $H$ is positively homogeneous of ...
0
votes
2answers
59 views

Find angles of triangle formed by images of vertices about opposite sides of an isosceles triangle

In $\triangle ABC$, $AB=AC$ and $\measuredangle BAC=30^\circ$. If $A^\prime$, $B^\prime$ and $C^\prime$ are the reflections of $A$, $B$, and $C$ about $BC$, $CA$ and $AB$, How to find $\measuredangle ...
0
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1answer
27 views

Shortest expression for the diagonal in a convex non-cyclic quadrilateral knowing its sides and the other diagonal?

I'm trying to arrive to the shortest expression possible for finding the diagonal in a convex (and non-cyclic) quadrilateral, knowing its four sides lengths and the other diagonal. My best try ...
2
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1answer
49 views

What does |AM|=|AC| mean?

What does $$|AM|=|AC|$$ mean? $$AM$$ and $$AC$$ are rays. Do they mean the length or what?
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0answers
21 views

Is there a converse for the equal intercept theorem?

If equal intercepts are made by transversals on three or more lines, then are the lines parallel to each other?
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1answer
28 views

Order of affine reflections (described with complex numbers operations)

Let be the affine reflection described as an operation with complex numbers : $$s_\beta,_\nu : z \mapsto \nu + \overline{\beta z},$$  where $z, \nu \in \Bbb C$ and $\beta \in \Bbb C^1 = \{x+iy \ ...
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vote
1answer
37 views

Cyclic quadrilateral of any rectangle

Is there proof that any rectangle is a cyclic quadrilateral? Context: in Euclidean geometry, a cyclic quadrilateral or inscribed quadrilateral is a quadrilateral whose vertices all lie on a single ...
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1answer
41 views

Geometry with circles.

Two circles, with centres O and P respectively, intersect at A and B. The extension of OB intersects the second sircle at C and the extension of PB intersects the first circle at D. A line through B ...
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votes
2answers
33 views

Is it possible to bound the hypotenuse given the perimeter of an orthogonal triangle?

I am working on Project Euler's problem 9, which needs you to calculate the product of a pythagorean triplet which sums to 1000. Therefore we have: $a < b < c$ $c^2=a^2+b^2$ $a+b+c=1000$ I ...
0
votes
1answer
19 views

Rate of change of area depending of the circumference of a circle

I want to find the general formula which gives the rate of change of the area with respect to the circumference. (Of a circle) I know that I can use differentiation formulas but I don't want to do it ...
0
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1answer
22 views

Calculate the distance from Plickford to Murbell

Attached is my question. Please provide an explanation for how I could calculate the distance from Plickford to Murbell.
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1answer
38 views

Find ratio of areas of triangle to pentagon?

ABCDE is a regular pentagon; rays AB and DC intersect at X. Now the area of triangle BCX is 1. What is the area of the pentagon? I figured out that the area of the pentagon is the square root of 5. ...
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votes
1answer
30 views

Find the magnitude of the angle

It's a nice problem and I have got a little bit and have some troubles.Could you please help me in finding the answer for the following problem? Let $ABCD$ be a convex quadrilateral such that $\hat{ ...
1
vote
1answer
26 views

Number of Pieces a regular $n$-gon is cut into by its diagonals [closed]

In how many pieces a regular n-gon is cut into by its diagonals? I need a general formula. By inspection, I have the solution to some lower values of $n$. For $n=3,4,5,6$ solutions are $1, 4, 11, ...
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vote
1answer
13 views

The complement of a union of uncountable many curves in $\mathbb{R^3}$

Let $C=\bigcup\limits_{i\in I}C_i$, where $I=[0,1)$ and $C_i$'s are curves in $\mathbb{R^3}$ such that there is only one intersection point for each pair $(C_i,C_j)$ if $i\not= j$ in $I$. Is it ...
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2answers
23 views

Proof that a secant line intersects a circle in exactly two points (according to Hilbert's axiomatic system)

With Hilbert's axiomatic system, How do I prove that a non-tangent line $d$ that intersects a circle $C$ intersects it in exactly two point? My teacher gave us the following clue: First show that if ...
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0answers
11 views

Is the angle between the most outer vertices bigger or smaller then $\pi$

I have multiple vertices with a common startpoint on a plane in $\Bbb R^3$, so the problem can be simplified to $\Bbb R^2$. I want to check if the angle between the two most outer vertices is bigger ...
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0answers
17 views

Finding rotation and translation of a planar object in 3d

I have a planar object, say a polygon $A_1A_2\ldots A_n$ in the 3-dimensional Euclidean space. It is translated by a vector $v$ and rotated by a rotation matrix $R$, and the resulting image is ...
0
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1answer
30 views

construction of line segment of a length $\sqrt{a^2-b^2+c^2+d^2}$

There are line segments $a, b, c, d$ and $a > b$. I have a question how to construct a line segment of a length $\sqrt{a^2-b^2+c^2+d^2}$. I can use Pythagoras theorem but I don't know how to make ...
2
votes
1answer
27 views

Tricky collinear vectors problem

Let OABC be any planar quadrilateral. Let $G_1, G_2$ and $G_3$ be the centroids of OAB, OBC and OAC respectively, and let G be the centroid of the triangle $G_1G_2G_3$. Show that the points O, G and ...
2
votes
2answers
46 views

Geometry problem involving orthocentre and midpoint of sides.

Let $AA_1, BB_1, CC_1$ be the altitudes of $\Delta ABC$ and let $AB \neq BC$. If $M$ is midpoint of $BC$, $H$ the orthocentre of $\Delta ABC$ and $D$ the intersection of $BC$ and $B_1C_1$, prove ...
2
votes
0answers
95 views

What is the best approximate of points on a sphere?

I have a unit radius sphere with a set $S$ of $n$ points on it. How can I find a map $f:S\to \mathbb{R}^4$ which minimizes $$\sum_{x,y\in S} \bigg( d_{\text{geodesic}} (x,y)^{2} - ...
2
votes
1answer
43 views

Is there a problem of plane geometry whose analytic reformulation gives a polynomial non-solvable by radicals?

This answer explains that any elementary plane geometry problem can be reduced to the existence of a solution of a polynomial system (called the analytic reformulation). Question: Is there a ...
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0answers
30 views

Geometry (using vectors or complex numbers) problem involving points on a triangle and on a circle

Let ATS be a fixed acute-angled triangle, i.e., all the three angles of the triangle are less than 90 degrees. Let E and F be two points on the sides AS and AT, respectively, such that the segment EF ...
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0answers
24 views

How to embed points on a sphere in a 3 or 4 dimensional space

I am looking for a procedure to embed points on a sphere in a 3 or 4 dimensional Euclidean space such that the distances are preserved as much as possible. If there is any related optimization ...
4
votes
3answers
122 views

Is it possible to project orthogonally an ellipse with major and minor axes $2a$,$2b$ so that its image is a circle with diameter $2b$?

Problem: Prove that the area of an ellipse with major axis and minor axis of lengths $2a$ and $2b$,respectively, is $ab \pi$ . Proof: We do this by projecting the ellipse into a figure whose ...
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0answers
37 views

Apostol's Mathematical Analysis: On the Geometric Representation of Real Numbers

In Apostol's Mathematical Analysis (second edition), it is written, on p.3: The real numbers are often represented geometrically as points on a line (called the real line or the real axis). A ...
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1answer
51 views

Condition for three lines to be concurrent.

A triangle $\bigtriangleup ABC$ is given, and let the external angle bisector of the angle $\angle A$ intersect the lines perpendicular to $BC$ and passing through $B$ and $C$ at the points $D$ and ...
0
votes
1answer
23 views

Cyclic hexagon with every other side equal

Let $ABCDEF$ be a cyclic hexagon with $AB=CD=EF$. Let $AC\cap BD=P, CE\cap DF=Q, EA\cap FB=R$. Prove that $\triangle PQR\sim\triangle BDF$. This problem seems simple, but I'm having trouble figuring ...
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3answers
58 views

Shown $p \in \mathbb{Z} [i]$ is a prime given $p\in \mathbb{Z}$ is a prime and $p$ does not equal $x^2 + y^2$

Suppose $p \in \mathbb{Z}$ is a prime number for which there are no integers, $x$, $y$ such that $p = x^2 +y^2$. How can I go about showing that $p$ is a prime element of $\mathbb{Z} [i]$. Assuming ...