# 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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### How do i compute the closest points on a sphere given a point outside the sphere?

I looking for method which can compute the yellow area in this image.. The ball with the green fill is a sphere, where i know the center point and the radius of it. The circle with the red fill ...
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### What is the best way to inscribe a golden rectangle into a pentagon? Do more golden ratios emerge?

Below I drew a golden rectangle in a pentagon in Adobe Illustrator. What would be the best way to inscribe a golden rectangle into a pentagon as shown in the figure below in a mathematical manner? ...
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### Distance between two lines

Find the distance between the lines $l_1:$ $x=1+4t,y=5-4t,z=-1+5t$ and $l_2:x=2+8t,y=4-3t,z=5+t$ So the approach in general is to find a vector that is orthogonal to 2 planes that the lines are in ...
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### 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 ...
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### Checking if vector crosses the simplex

Let assume that I have a point in $x \in \mathbb{R}^n$ Also I have a non-zero vector defined by it's endpoint attached to this point. The third thing I have is a simplex of $\dim=n$, such that the ...
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### Show that if a convex polygon is entirely inside another convex polygon, then the outer polygon has perimeter greater than or equal to the inner one.

Can anyone help me out with proving this statment? "Show that if a convex polygon is entirely inside another convex polygon, then the outer polygon has perimeter greater than or equal to the inner ...
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### Proof/justification that a circumscribed regular polygon has a perimeter greater than the circumference of the circle?

According to Archimedes, the perimeter of any circumscribed regular polygon is greater than the circumference of the circle. ie: http://www.themathpage.com/atrig/Trig_IMG/eval1.gif This does seem ...
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### How can I find ratio between area of triangle and area of quadrilateral?

I have a parallelogram $ABCD$. $E$ is center of $AD$. $O$ is center of $AC$ and center of $DB$. $F$ is the intersection point between $CE$ and $DO$. Point $G$ is the intersection between $EO$ and $AF$....
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### Minimum Perimeter of a triangle

I have been playing the app Euclidea, I have been doing quite well but this one has me stumped. "Construct a triangle whose perimeter is the minimum possible whose vertices lie on two side of the ...
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### Defining perpendicular lines in the 3D space

Is there a universal agreement about the definition of "perpendicularity" between two stright lines in the 3 dimensional euclidean space? Do they need to meet or it is enough to have perpendicular ...
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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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### Prove that if $A \perp B$ and $B \perp C$, then $A \parallel C$

Suppose that $A$, $B$ and $C$ are non-zero vectors in $\mathbb R^2$. Show that if $A$ and $B$ are orthogonal and $B$ and $C$ are orthogonal then $A$ and $C$ are parallel. I feel like this should be ...
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### Online tool for making Geometric Constructions.

There was a website where it tasked you making different geometric shapes using only a compass and straightedge. I've looked for it and I can't find it or even discussion about it. What I do remember ...
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### General conic equation and coefficient matrices

For a general conic $Q(x,y)=ax^2+2hxy+by^2+2gx+2fy+c$ we define a matrix $A$ as follows: $A=\left( \begin{matrix} a& h& g\\ h& b& f\\ g& f& c\end{matrix} \right)$. Then we ...
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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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### 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: ...
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### 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 ...
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### 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 ...
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### 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}} \...