# 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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### 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 ...
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### 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 <...
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### Moving Line Segment Problem part 2

This question is related to a question I asked a while ago here on math.stackexchange: Moving Line Segment Problem The rules for how the line segment can be moved are the same: The endpoints must ...
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### Isogonal Conjugate of point outside of triangle

I was wondering about reflections of lines over the external bisectors instead of external bisectors in a triangle. Here is a problem that brought it up: Let $P$ be a given point inside quadrilateral ...
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I have a simple question about the formula for the volume of the cone. Let $C$ a cone, which base has radius $r$ and height equal to $h$. So its volume can be compute by the formula: $$\text{Vol}(C)=\... 1answer 32 views ### Prove perpendicular bisectors of non-parallel lines intersect Suppose that A, B and C are points and that AB and BC are not parallel. Show that the perpendicular bisector of AB, l, and the perpendicular bisector of BC, l', are not parallel and ... 0answers 22 views ### Prove Concurrency using Radical Axis of Circumcircles Let the incircle of \triangle ABC touch sides BC,CA,AB at D,E,F, respectively. Let \omega,\omega_1,\omega_2,\omega_3 be the circumcircles of \triangle ABCm,\triangle AEF,\triangle BDF,\... 2answers 340 views ### Optimal path around an invisible wall [duplicate] The Problem On an infinite plane there are two points, A and B, a unit distance apart. There is a 50\% probability that there is an invisible wall somewhere between the two points. The ... 1answer 16 views ### Characterizing isotropic measures A Borel measure \mu on S^{n-1} is called isotropic if$$\int_{S^{n-1}} \langle \theta, x \rangle^2 d\mu(x)=\frac{\mu\left({S^{n-1}}\right)}{n}$$for all \theta\in S^{n-1}. This means that in ... 3answers 80 views ### Axioms of Geometry? I have taken the generic low level undergraduate classes, such as Calculus 1-3, Differential Equations, and Linear algebra. Since I never learned Geometry past a basic high school level, I thought it ... 2answers 27 views ### Prove that DE || BC Let M be the midpoint of side BC in triangle ABC. The angle bisector of BMA intersects AB in D, while the angle bisector of CMA intersects AC in E. How can i prove that DE||BC? I drew out the ... 1answer 39 views ### Finding the set of points on the sphere with an equal product of distances Given two points x_1 and x_2 on the sphere, one can find another set of points on the sphere \{y_1, y_2\} such that the product of Euclidean distances to the given points x_i is the same for ... 0answers 27 views ### How many Pascal hexagons can I construct with 6 different points on a circle? I have a basic knowledge about combinatorics and I am in a euclidean geometry class. My question is : How many Pascal hexagons can I construct with 6 different points on a circumference? It could ... 1answer 38 views ### Name of the geometric figure of points {\bf x} \in \Bbb R^n with 1-norm ||{\bf x}||_1 = 1 Is there a name for the figure$$\{{\bf x} \in \Bbb R^n : ||{\bf x}||_1 = 1\} \subset \Bbb R^n ?$$Things like this seem to usually have names, for instance, the n-cube or n-ball. In 2 ... 5answers 69 views ### what is the value of angle A The triangle ABC is random. The line AD is twice big as the line DC (AD=2*DC). We know only the two angles that are shown in the picture. What's the value of angle A? 1answer 31 views ### Median BM of triangle ABC two results Given Calculate the measure of the median \overline{BM} of ABC triangle, given A (-6.1); B (-5,7) and C (2,5) I get this result: Xm = \frac{Xc - Xa}{2} + Xa Xm = \frac{2-(-6)}{2} + (-6) = ... 11answers 5k views ### In a right triangle, can a+b=c? I understand that due to the Pythagorean Theorem, a^2+b^2=c^2, given that a and b are legs of a right triangle and c is the hypotenuse of the same right triangle. However, most of the time, a+... 1answer 37 views ### Is f(x) = x^2 a scalar function? Take a simple parabola. It is a function that has a one-dimensional co-domain$$f(x) = x^2 $$It is mapping the set of values in its domain, to one-dimensional values in its co-domain, and it ... 1answer 69 views ### Why this function describes a euclidean ball? In Stephen Boyd's convex optimization book at page 97, one can read :$$ a,b \in R^n  (1-\alpha^2)x^Tx-2(a-\alpha^2b)^Tx+a^Ta-\alpha^2b^Tb \leq 0 $$is convex (in fact a euclidian ball) if  \... 1answer 24 views ### Find distance of overlapping squares How to find distance center to center from square 1 to square 3, if we need overlap area is 15.46 mm^2. if we know each side of the square is 6.9 mm. Firstly I find the distance is 9.3 mm ... 0answers 15 views ### Bounds on the size of Voronoi cells I am working on an algorithm for which bounds on the size of voronoi cells will come in handy. Suppose that the domain D is partitioned according to the Voronoi cells D_1,\dots,D_n with Voronoi ... 1answer 18 views ### Perturbation of tangent ball As picture below, A and B are two balls, \partial A\bigcap \partial B=\{k\}, and B contains A. How to show that$$ \forall h\in \partial B,\exists ~\varepsilon > 0 ~st~ A\subset B+\...
Pappus's hexagon theorem: Given one set of collinear points $A,B,C$, and another set of collinear points $a,b,c$, then the intersection points $X,Y,Z$ of line pairs $Ab$ and $aB$, $Ac$ and $aC,Bc$ and ...