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
92 views
How to determine oriented angle of rotation in 3-dimensional space
Let $V$ be oriented two-dimensional Euclidean space. Then we can define an oriented angle $\phi$ between two nonzero vectors $u,v\in X$ by formulas: $ \phi=\arccos \frac{\langle u,v\rangle}{\|u\| ...
3
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3answers
4k views
Check if a point is within an ellipse
I have an ellipse centered at $(h,k)$, with semi-major axis $r_x$, semi-minor axis $r_y$, both aligned with the Cartesian plane.
How do I determine if a point $(x,y)$ is within the area bounded by ...
0
votes
4answers
135 views
Figure out if a fourth point resides within an angle created by three other points
If I have a point that is considered the origin and two lines that extend outwards infinitely to two other points, what is the best way to determine whether or not a fourth point resides on or within ...
1
vote
2answers
174 views
Triangle in hexagon
In a regular hexagon ABCDEF is the midpoint (G)of the sides FE and S
intersection of lines AC and GB.
(a) What is the relationship shared point of straight ...
0
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2answers
62 views
Faster Distance formulae for higher n dimension
I need to calculate the distance two points, but these two points contain more than 100 dimensions. With the regular two or three dimension distance formula, we can extend this further to n dimension ...
1
vote
2answers
50 views
Determine whether the triangles $ABC$ and $DEF$ are rectangles
How can we determine whether the triangles $ABC$ and $DEF$ are rectangles? We have $A(-6,5),B(-3,3),C(1,9),D(1,3),E(5,1),F(11,10)$.
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vote
2answers
86 views
finding area of the fourth circle
Three circles of the same radius are arranged in such way that one circle is tangent to the other two. A fourth circle is drawn so that it will contain three circles and be tangent to the other ...
1
vote
2answers
282 views
A geometry problem from maths competition
I have a problem understanding the proof.
Given an acute angle $A$. Choose an arbitrary point $P$ from the bisector of $A$ and another point $B$ from the side of angle $A$. Draw a line $l$ going ...
2
votes
1answer
105 views
Prove that the Simson line of $P$ bisects the segment $HP$ from the orthocentre $H$ to $P$
Let $ABC$ be a triangle with orthocentre $H$ and circumcircle $\odot(ABC)$.
Suppose $P\in\odot(ABC)$. Let $\gamma$ be Simson's line of $P$ wrt $ABC$.
Prove that $\gamma$ bisects $PH$.
5
votes
1answer
256 views
Find volume of crossed cylinders without calculus.
I found this puzzle here. (It's labeled "crossed cylinders".) Here's the description:
Two cylinders of equal radius are intersected at right angles as shown at left. Find the volume of the ...
1
vote
2answers
95 views
area of a circle - 3/4th
How to find the pixels of that line which is crossing the circle?
Is there any formula?
Iam getting the line's end points
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votes
1answer
48 views
Euclidean Conjugation group
Having a tad bit trying to prove this question,
Show that the set of reflections is a complete conjugacy class in the euclidean group E. Also, do the same for the set of half-turns and inversions.
...
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votes
3answers
269 views
Is there a uniform way to define angle bisectors using vectors?
Look at the left figure. $x_1$ and $x_2$ are two vectors with the same length (norm). Then $x_1+x_2$ is along the bisector of the angle subtended by $x_1$ and $x_2$. But look at the upper right ...
8
votes
0answers
123 views
Who first discovered that the torus supports a flat structure?
Who first recognized that there exists a homogenous metric on the closed genus 1 orientable surface?
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votes
1answer
31 views
Determining the vectors of a tetrahedron
Suppose that a regular tetrahedron with vertices $A$, $B$, $C$, and $D$ has its centroid at the origin $O$, as in the below schematic. Vectors $OA$, $OB$, $OC$, and $OD$ each have length $\ell$ ($|OA| ...
1
vote
4answers
124 views
Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1
How would I go about proving this: Prove that if a, b, x, y are integers with ax + by = gcd(a, b) then gcd(x,y)= 1
2
votes
2answers
153 views
Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle
How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
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votes
2answers
464 views
Solid angle between vectors in n-dimensional space
There is a formula of to calculate the angle between two normalized vectors:
$$\alpha=\arccos \frac {\vec{a} \cdot\ \vec{b}} {|\vec {a}||\vec {b}|}.$$
The formula of 3D solid angle between three ...
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vote
2answers
66 views
Maximal area of triangle if angle and opposite side length is known
Two lines $l_1$ and $l_2$ intersects at point $A$ such that the angle they intersect is $\alpha$. A line segment has endpoints $B$ and $C$ in the lines $l_1$ and $l_2$, respectively, and $|BC|=l$. ...
0
votes
0answers
92 views
Calculate coordinates of the a point in space with hypotenuse and two angles given
I have a cylinder with a length of $2$, and two angles for rotation around two of the axes. Functions for that are named $\text{RotX}$ (rotation around X axis) and $\text{RotZ}$ (rotation around Z ...
1
vote
1answer
170 views
Projection of 5 skew lines
Given five skew lines, is it possible to find a point $P$ and a plane $\pi$ such that the projections of the five lines from $P$ onto $\pi$ intersect in the same point $Q$? [editet: rewritten clearly, ...
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vote
1answer
106 views
Maximum possible area of triangle PQR
A point $P$ is given on the circumference of a circle of radius $r$.Chords $QR$ are drawn parallel to the tangent at $P$.Then how can we determine the largest possible area of triangle $PQR$?
Thanks.
...
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votes
1answer
92 views
there are two docks
There are two docks, dock A and Dock B, on a large lake. The distance between the two docks is 72.5 km. Dock B is directly east of dock A. One day, a steam boat leaves from dock A at noon, and heads ...
2
votes
3answers
110 views
Sides of triangle and an altitude
Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Let $h$ be the altitude drawn on the side of length $a$ Then is $a^2 + 4h^2 - (b+c)^2$ always negative ?
3
votes
2answers
88 views
Construct tangent to a circle
Using a ruler and a compass how can construct a line through a point and tangent to a circle. What I don't want is to eyeball the line by trying to line-up the ruler over the circle. Best if I could ...
1
vote
1answer
175 views
Relationship between the sides of inscribed polygons
In my math textbook there's a demonstration for the calculus of the circumference of a circle that involves regular polygons inscribed in the circle, but I don't get it. The book gives the following ...
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votes
2answers
20 views
Possible constraints on weight constraints for a point position
Suppose a line is given in 2D, and a set of $k$ arbitrary points $x_1, x_2, \dots, x_k$ along that line. For some non-zero weights $w_i$ associated with each $x_1$, it can be shown that a weighted ...
3
votes
4answers
1k views
Find the area of overlap of two triangles
Suppose we are given two triangles $ABC$ and $DEF$. We can assume nothing about them other than that they are in the same plane. The triangles may or may not overlap. I want to algorithmically ...
3
votes
2answers
95 views
Is every triangle a quadrilateral?
I can imagine a quadrilateral where one of the angles is $180^\circ$. Is this still considered a quadrilateral?
More generally, is every $n$-gon also a $(n+1)$-gon (for $n \ge 3$)?
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votes
2answers
129 views
Reading geometry problems.
Prove that the external bisector of an angle of a triangle (not isosceles) divides the opposite side (externally) into two segments proportional to the sides of the triangle adjacent to the angle
...
3
votes
0answers
106 views
Menelaus's Theorem clarification
If three points, one on each side of a triangle (extended if necessary) are collinear, then the product of the ratios of division of the sides by the points is -1 if internal ratios are considered ...
3
votes
1answer
407 views
Can two planes intersect in a point?
Is it true that two planes may intersect in a point ?
or
If they intersect then, they always make a straight line ?
I have some doubt; please explain.
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votes
2answers
130 views
line equidistant from two sets in the plane
Suppose $A$ and $B$ are disjoint subsets of the plane, both closed, nonempty, and connected. Define $E(A, B)$ as the set of points in the plane equidistant from $A$ and $B$. For example, if $A$ is a ...
2
votes
0answers
65 views
What is requirement to inscibe sphere in pyramid with quadrilateral base?
In certain math problem the only information about pyramid with quadrilateral base is that you can inscribe sphere inside. What kind of constraints was put on my pyramid ABCDS?
Is it something ...
3
votes
3answers
731 views
Finding the area of a quadrilateral
I have a quadrilateral whose four sides are given
2341276, 34374833, 18278172, 17267343 units.
How can I find out its area? What would be the area?
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votes
1answer
66 views
Is this circular logic on geometry proof?
I am trying to prove that the internal bisectors of the angles of a triangle meet at a point - the incenter.
I need someone to critique this incomplete proof for me
Consider $\triangle ABC$ ...
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votes
0answers
32 views
Is this geometry statement incomplete?
If a line intersecting two other lines makes the alternate angles equal to one another, then the lines will be parallel.
Now if you go to this neat website
...
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vote
0answers
16 views
3D orientations from distance constraints
I want to determine the relative orientations within a set of rigid 3D objects given some pairwise distances between certain points on pairs of objects. There are sufficient constrains to fully ...
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vote
3answers
116 views
Number of ellipses through two fixed points in 2D space?
How many ellipses with a given size (mean $a$ and $b$ given) one can draw through two fixed points in 2D plane?
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votes
1answer
40 views
Reason for hardness of optimal minimization, and use of iterative optimizers
Suppose a set of $n-1$ are given in 2D space, $x_1, x_2, \dots, x_{n-1}$, and an additional point $x_n$ is to be assigned a 2D coordinate such that the prescribed Euclidean distances $d_1, d_2, \dots, ...
0
votes
1answer
397 views
How do I determine the dihedral angle of two (bond-defined) planes?
I'm not sure if this question would be more appropriate for Chemistry or Physics SE, but if so please forgive me.
I have drawn the following picture; the spheres represent atoms and the lines ...
0
votes
0answers
96 views
Maximum diameter of a 2D shape
What is the diameter of an arbitrary 2D figure? (Diameter=The longest distance between two points within the 2D figure). What is the most efficient algorithm? Is it an exact one? Specifically, could ...
5
votes
0answers
100 views
Hidden geometrical gems in Euclid's Elements?
I am teaching a course in Euclidean geometry at the University of South Carolina, and it seemed highly appropriate and interesting to read Euclid himself. (See here for a wonderful, and completely ...
6
votes
1answer
95 views
Proving $R\ge 2 r$ using synthetic geometry
If $R$ and $r$ be the radii of the circumcircle and incircle of a triangle, then how do I prove by synthetic geometry(i.e. without trigonometry) that $R\ge 2r$?
I am aware of a trigonometric proof ...
3
votes
2answers
342 views
Average Distance Between Random Points in a Rectangle
My question is similar to this one but for rectangles instead of lines.
Suppose I have a rectangle with sides of length $L_w$ and $L_h$. What is the average distance between two uniformly-distributed ...
2
votes
0answers
33 views
Euclidian embedding of lines
I'm looking for a way to convert a set of lines in R^3 into points in R^n so that distance between any pair of points points is a good approximation of the distance between corresponding pair of ...
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votes
2answers
74 views
Rectangular problem
I was trying to solve this problem:
Let P be a point in the interior of rectangle ABCD. Given PA = 3, PD = 4 and PC = 5, find PB.
I feel lost because it's not right to assume P is in the center ...
1
vote
1answer
88 views
Relation: average (squared) distance to all points, and (squared) distance to centroid
Suppose a set of $n$ high-dimensional points is given. It is known that the sum of all pair-wise squared Euclidean distances is proportional to sum of squared distances of all points to the centroid.
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votes
1answer
81 views
Finding the x- coordinate in triangle
Is it possible to find the point which is marked by question mark ? we know that the s1(x)=s2(x) (the areas of the two triangles are equal)
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2answers
152 views
Cutting the corners of a cube
Do you know of any way to cut the corners of a cube by means of rotation assuming that the cube is centered in the origin of XoYZ?
For example if we have a square centered in the origin of XoY and we ...
