# Tagged Questions

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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### Found a New Golden Ratio Construction with Equilateral Triangle, Square, and Circle. Geometric/Trigonmetric proof?

The below figure discloses a new golden ratio construction with an equilateral triangle, square, and circle. Geogebra gave me the value of the golden number 1.618 for the ratio of the yellow line to ...
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### USAMO 2005, Problem3 (Triangle Geometry)- Is my solution correct?

USAMO 2005, Problem 3: Let $ABC$ be an acute-angled triangle, and let $P$ and $Q$ be two points on its side $BC$. Construct a point $C_{1}$ in such a way that the convex quadrilateral $APBC_{1}$ is ...
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### Banach-Mazur distance from finite-dimensional subspaces of $\ell_p$ to the Hilbert space

I am reading a paper http://www.math.tamu.edu/~johnson/TF3.4.pdf by Bill Johnson and Andrzej Szankowski and having trouble grasping why $d_n(Z_m) \leq d_n(\ell_{p_{m+1}} ) = n^{|p_{m+1}-2|}$ in the ...
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### Geometric inequality $180^{\circ}\left(1-\frac1n\right)\le AMB$

Point $M$ is located inside a regular $n$-gon. Prove that there exist different vertices $A$ and $B$ that $$180^{\circ}\left(1-\frac1n\right)\le AMB\le 180^{\circ}$$ My work so far: Let ...
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### New Golden Ratio Construct: which one of my constructs is superior/simplest--squares & circles or just circles?

I have found yet another golden ratio construction. Geogebra gives it the value of 1.61803398874990 to the ratio between the yellow and blue lines in the figure below, which is the golden ratio PHI. ...
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### maximal volume/diameter of polyhedron

I am trying to develop an integral in $R^n$ and I am faced with the following problem: Given a polyhedron $P$ in $R^n$ of diameter d, define the "compactness" of $P$ as the quotient of the volume of ...
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### The Golden Ratio in a Circle and Equilateral Triangle. Geomertic/Trigonometric Proof?

Geogebra gave me 1.61 for the following Golden Ratio construction shown below. Firstoff, has anyone seen anything similar to this construction? Basically begin with an equilateral triangle. ...
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### An interesting geometry problem with angle bisectors and tangent

I have found the following problem: There is an acute $\triangle ABC$. Denote its circumcircle as $\omega$. The angle bisector of $\angle BAC$ intersects $BC$ and $\omega$ in points respectively $A_1$ ...
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### Distance between incentre and orthocentre.

I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C}$$here $r$ is inradius and $R$ is circumradius. I considered $\triangle API$ ($P$ is ...
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Verifying that (p=2) satisfies $$\forall n\in\mathbb{Z}^+.\exists A\in(\mathbb{R}^{n\times n}\setminus\{I_n\}).\forall k\in\mathbb{R}.\forall ... 1answer 51 views ### Ellipse's farthest point to another point I am trying to find the farthest and closest points of a ellipse without using any brute force type of coding. The processing power is limited so it should be as pinpoint as possible. I have tried a ... 1answer 36 views ### Prove any line passes through at least two points I've started reading Introduction to Algebra by Cameron, and I'm stuck on the first exercise. Q. Prove any line passes through at least two points using the axioms given below. Definitions: ... 2answers 46 views ### Assuming that the sum of the angles of any triangle is 180, how can I deduce Euclid's 5th postulate? I already did the reverse, namely, if we assume Euclid's 5th postulate, then the sum of the angles of any triangle is 180 degrees. Now I need to show the converse, but I don't really know how to ... 0answers 35 views ### The Golden Ratio in a Circle, Triangle, and Square: simple geometry/trigonometry construction. I believe that I have found the golden ratio in the below figure. Geogebra is saying it is very close. Might anyone have a geometric/trigonometric proof? An equilateral triangle ABC is inscribed in a ... 2answers 27 views ### Confusion on wording of an elementary geometry problem I really want to know the following geometry problem is valid or not. (Please don't change the wording of the problem. Please answer it is valid or not. Please answer frankly.) "ABCD is a ... 2answers 40 views ### Power of a point proof I found the question on page 13 of this link. Let P be a point inside a circle such that there exist three chords through P of equal length. Prove that P is the center of the circle. I ... 0answers 26 views ### How to find all those points whose distance from x=(2,0) is minimum, using \|x\|=|x_1| + |x_2|? The points must be in the closed ball \{y : \|y\| \le 2\|x\|\}. I know |y_1|+|y_2| needs to be \le 4. Other than that, I am confused about how to find all the points that are minimum distance ... 0answers 7 views ### Rewrite each isometry as the composition of at most three reflections Write each of the following isometries as a composition of at most three reflections: \rho_{(1,0),\frac{\pi}{6}} \tau_{(1,0)+(0,1)}=\tau_{(1,1)} \sigma_{l_{BC}} \rho_{(1,0),\frac{\pi}{6}} ... 0answers 7 views ### Prove that l and l_{AB} are parallel if and only if \sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id Prove that l and l_{AB} are parallel if and only if \sigma_B \sigma_l \sigma_B \sigma_A \sigma_l \sigma_A = id I imagine that this proof has to be along the lines of a proof by contradiction, ... 3answers 123 views ### Can a figure inside a circle be seen at right angle from any point on the circle? A convex, closed figure lies inside a given circle. The figure is seen from every point of the circumference at a right angle (that is, the two rays drawn from the point and supporting the convex ... 1answer 493 views ### Incentre and excentre of a triangle Prove that the triangle formed by the points of contact of the sides of a given triangle with the excircles corresponding to these sides is equivalent to the triangle formed by the points of contact ... 1answer 417 views ### Properties of sphere Let C be a circle with diameter \overline{AB}. Then it is well known that for any P on the circle C the angle \angle APB =\frac \pi 2. There are similar results for sphere? 0answers 23 views ### To find/create midpoint, is it easier to bisect a line segment, or double a line segment? With only compass and ruler / straightedge. [on hold] Suppose one wants to find the midpoint of a line segment. Is it generally easier to simply draw two lines of equal length end to end, or is it easier (does it count as less steps) to draw a line and ... 1answer 36 views ### Simple Golden Ratio Construction with Three Lines, and Interesting Implied Circle? Consider three equal lines (as illustrated below). A red, green, and orange line of equal length all rest upon the same horizontal line. The red line is stood upon its end in a manner perpendicular ... 4answers 46 views ### Golden Ratio Conjecture in three simple Geogebra shapes--circle, triangle, and square. A circle, equilateral triangle, and square of equal heights are all placed on the same horizontal line as shown below. The circle is tangent to the triangle which is centered upon the left edge of ... 0answers 26 views ### PHI Conjecture: Fibonacci Number Line Segments & Simple Golden Ratio Puzzle? This is a golden ratio conjecture. Does the golden ratio exist as conjectured below? The three line segments in the figure below have lengths of the Fibonacci numbers 2, 3, and 5. blue=2 green=3 ... 1answer 23 views ### Angles in Hilbert's axioms for geometry In Hilbert's axioms for geometry, the following elements are presented as undefined (meaning "to be defined in a specific model"): point, line, incidence, betweenness, congruence. In deed, when ... 1answer 62 views ### Two Equilateral Triangles and the Golden Ratio: Simple Geomtric Proof Two equilateral triangles rest upon the same horizontal line, as shown in the linked figure below. The red triangle is tilted so that one corner touches a side of the black triangle at the midpoint ... 1answer 19 views ### Is this definition of a Euclidean frame well-defined? Going through my lecture notes on geometry I find a definition of a Euclidean frame which doesn't seem to have been formed correctly (most likely written down wrong). So I've taken it upon myself to ... 1answer 35 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, ... 1answer 34 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 ... 1answer 17 views ### How to fit a convex quadrilateral inside another short of cutting them out and playing with them? I have two convex quadrilaterals (ABCD and WXYZ). Their sides and their interior angles are known. I also know that WXYZ fits perfectly inside ABCD with each corner point touching a different side. ... 1answer 185 views ### Characterization of linearity in terms of metric At least in Euclidean geometry and the upper half plane model of hyperbolic geometry, the statements 'y lies on the line segment determined by x and z ' and 'd(x,y)+d(y,z)=d(x,z) ' are ... 0answers 86 views ### Water filling problem in Blocks - Algebra Question Consider a rectangular plot comprising n\times m square cells on which nm cement blocks of various heights are stored, one block per cell. The base of each block covers one cell completely, and ... 0answers 31 views ### Finding a relation between three points in a small circle of a sphere I have a relation as follows. I am given two points C, D on a circle, and a point P somewhere inside of it. I would like to find the quantity |AC| |AD|, where A is a point on the circle lying ... 3answers 58 views ### A simple geometrical question regarding three circles and a line. Trigonometric construction. [closed] In Figure 1 three tangential circles all have the radius of 1 or r. What is the ratio of the blue line to the yellow line in terms of r, and in terms of r=1? 1answer 19 views ### Isometries in \mathbb{E}^2 We define \mathbb{E}^2 = (\mathbb{R}^2, d_E), where d_E is the usual Euclidean metric on \mathbb{R}^2. We say that a function f:X \rightarrow Y, where X, Y are metric spaces, is an isometry ... 1answer 25 views ### An elementary problem in Euclidean geometry Let ABC be an acute triangle (AB < AC) which is circumscribed by a circle with center O. BE and CF are two altitudes and H is the orthocenter of the triangle. Let M be the ... 1answer 60 views ### Draw A Triangle From 3 Excenters and Ex-radii My teacher gave me this problem and told me to think- " Is it possible to draw a triangle, given the three ex-centers and length of the ex-radii?" I don't know if it's possible or not. So, my ... 1answer 62 views ### How to prove a regular pentagon is formed by knotting a rectangular strip of paper? I found this interesting problem from a friend (From Arthur Engel's-Problem Solving Strategies book). The method to begin the problem is as follows- Step 1.Take a rectangular strip of paper ... 1answer 57 views ### How to show \partial A = \varnothing \Rightarrow A=R^n [closed] Let A\subset R^n and dimA=n, \partial A is the relative boundary of A. If \partial A=\varnothing how to show A is R^n ? Picture below is from XX page of Schneider R.-Convex Bodies_ ... 2answers 551 views ### a conjugate of a glide reflection by any isometry of the plane is again a glide reflection Q : Prove that a conjugate of a glide reflection by any isometry of the plane is again a glide reflection, and show that the glide vectors have the same length. I know that: the conjugate ... 2answers 25 views ### Prove ∠ADM = ∠ACB of triangle ABC [closed] Suppose that ABC is a triangle. Let D be its circumcenter and let M be the midpoint of \vec {AB}. Show that ∠ADM = ∠ACB. 1answer 25 views ### Every orthogonal matrix represents a rotation around an axis Is it true that every element of the group O(n) represents a rotation around some axis? I'd like this to be true in order to decompose any matrix R \in O(n) as a block matrix in O(n-1) and a 1 ... 1answer 23 views ### There are finitely many n-dimensional wallpaper groups There are 7 Frieze groups, 17 wallpaper groups and 230 space groups, which are the 1,2,3-dimensional cases of isometries on \Bbb R^n (I think? Frieze groups seem to require [0,1]\times\Bbb R ... 0answers 9 views ### How to determine the angle from a point and the plane tangent points in a sphere I have an UAV modeled in three dimensions with let's say position coordinates p_{uav} = (x_1,y_1,z_1) that is moving in a direction d = (d_x,d_y,d_z) and a moving obstacle modeled as a sphere with ... 11answers 5k views ### In a right triangle, can a+b=c? I understand that due to the Pythagorean Theorem, a^2+b^2=c^2, given that a and b are legs of a right triangle and c is the hypotenuse of the same right triangle. However, most of the time, ... 0answers 27 views ### A geometric inequality about the internal besectors [closed] prove that in every triangle the following inequality is hold:$$\frac{1}{w_\alpha}+\frac{1}{w_\beta}+\frac{1}{w_\gamma}\le \frac{2}{\sqrt{3}}\left(\frac{1}{a}+\frac{1}{b}+\frac{1}{c}\right) ...
The Problem On an infinite plane there are two points, $A$ and $B$, a unit distance apart. There is a $50\%$ probability that there is an invisible wall somewhere between the two points. The ...