# Questions tagged [geometry]

For questions about geometric shapes, congruences, similarities, transformations, as well as the properties of classes of figures, points, lines, and angles.

31,926 questions
14 views

### Overlapping circles covering polygon

While working in GeoGebra I noticed something odd. I had a triangle with a point inside and the point was connected to each of the vertices. For each vertice I had drawn the circle passing through the ...
15 views

### Distance between a point and low-dimensional sphere

Is there a way to analytically calculate the distance between an arbitrary point $\mathbf{x}\in\mathbb{R}^n$ and a low-dimensional sphere embedded in $\mathbb{R}^n$, say one aligned with the axis ? ...
87 views

### $\mathbb CP^1 \approx S^2$ proof check

I wanted to give a whole proof of this fact as I was not able to find a detailed one myself. I have the feeling that such a proof has been asked quite frequently by several users and I hope this may ...
40 views

### How many points are needed to define a circumference?

This doubt comes from a combinatorics problem in a textbook, which states: Consider two strictly parallel lines and seven dots, four of which are over one of them, and three over the other. Three ...
22 views

### How would be a formal answer for an automata geometry problem?

Let an Automata A sitting on a point O (0,0) and turned to North. That Automata can execute only any combination of three different commands in each step: Move one unit forward Turn 90 degrees ...
34 views

### Are there always two circles that together surround or intersect all points in the following scenario?

Consider $N$ points in $\mathbb{R}^2$ and $\binom{N}{2}$ circles, one for each pair of points such that it intersects both. Is it always possible to pick two of these circles that together surround or ...
617 views

### Magnifying glass in hyperbolic space

My grandmother used to read with a magnifying glass. What (an ideal) magnifying glass does, is basically a homothety: it scales the picture by some factor. Now, in a hyperbolic space there is no such ...
714 views

### Rectangle inscribed in a circular sector of angle 60

My apologies if this has been asked before. Given a circular sector, say of radius $r$, with internal angle $60^{\circ}$, construct a rectangle inscribed in that sector so that the length of the ...
18 views

31 views

### Pair of lines problem

If the pair of straight lines $x^2+2xy+ay^2$ & $ax^2+2xy+y^2$ have exactly one line in common, then the combined equation of the two lines is given by A. $3x^2+8xy-3y^2$ B. $3x^2+10xy+3y^2$ C. ...
13 views

### How to calculate vertex, focus, axis etc. from such type of ellipse equation 3x²+8y²+12xy-18x-32y+23=0? [on hold]

Is there any way to find vertex , focus, axis, centre etc from this type of non ideal ellipse, hyperbola or parabola?
961 views

### Distances to line passing through the centroid of triangle

Let $p$ be a line that pass through the centroid of a triangle $ABC$. Unless the line pass through one vertex, then $2$ verices are one side of the line, while the third one is on the other side. ...
45 views

### Finding the lengths of $AC$ and $AD$

Triangle $ABC$ is right-angled at $A$. The angle bisector from $A$ meets $BC$ at $D$. If $CD=1$ and $BD=AD+1$, find the lengths of $AC$ and $AD$. I have tried to set up a equation with $AD$ and $AC$ ...
22 views

### A problem on combinatorical geometry . [on hold]

Hey could someone help with this exercice, i have tried everything but nothing seems to suite work. Any help would br appreciated : Let ABCDEF be a convex 6 sided polygon of sides 1, prove that at ...
39 views

### Geometry, does this shape have a name?

A Sphere with Diameter 1 perfectly inscribed in A cube with sides of 1, Removing the sphere and splitting the cube on the faces we then have 8 identical "Corners" with three sides being a tetrahedron ...
41 views

### What is the Geometric meaning of vector norm in Rn n>3

My question is related to the length of the vector , Sorry it may seem stupid for you as i come from engineering background not mathematics background For Vectors up to 3 dimensions (can be ...
42 views

### $|AB|+|BC|=l$ , find the position of $A,B,C$ maximize area of quadrilateral $OABC$ ？ [on hold]

The given condition is $|AB|+|BC|=l$ In a 2D coordinate system, point $O(0,0)$ is origin. With the following points: $A(0,a),a\geq0$ $B(b_1,b_2), b_1\geq0,b_2\geq0$ $C(c,0), c\geq0$ and the ...
10 views

### Chord of contact in polar coordinate

Can you please help me to derive the equation of chord of contact for a circle in polar coordinate? I have found the equation in cartesian coordinate but I cannot map that into the polar coordinate. ...
476 views

### Geometry proof problem (high school)

I have an upcoming chapter test and this was one of the practice problems. Can someone guide me? Given: Isosceles $\triangle ABC$ with $AB$ congruent to $AC$; $AD$ is not a median of $\triangle ABC$...
18 views

### Möbius Transformation on Riemann Sphere

Just started learning about Möbius Transformations myself, and I wanted to know what kind of transformation on the Riemann Sphere would preserve $S^1$ (unit circle) as a set? What would the conditions ...
24 views

### Showing complex FLT to be continous

If I have the complex Mobius Transformation, $$F_X (z) = \frac {az + b}{cz + d}$$ how can I show that the transformation is continous w.r.t the metric on $\Bbb C \cup \infty$ which is ...
14 views

### area of shaded region in sqm when the area of each circle is 2 sqm?and there are four circles [on hold]

The answer should have $\pi$ just like the options
40 views

### If P(Q)R and Q(R)S, prove P(Q)S

$$\text{If }\ P(Q)R\ \text{ and }\ Q(R)S,\; prove\ P(Q)S$$ Axioms: (i) If P(Q)R, then R(Q)P (ii) If P(Q)R, then $\sim$(P(R)Q) and P $\neq$ Q (iii) There are at least two distinct ...
8 views

### Affix of a point [on hold]

Let $A(a)$, $B(b)$ and $E(1)$ three points of the unit circle $\mathbb{U}$. Let $P(p)$ the point of $(AB)$ so that $(AB)\perp (EP)$. Find the affix $p$ of the point P.
12 views

### Stereographic Projection question

I want to check if stereographic projection formula, $$\Omega (x,y,z) = (\frac {x}{1-z}) + (\frac {y}{1-z})i$$ matches the following description: We can identify the complex plane with a ...
676 views

### Finding the area of a triangle when only a the shaded part and two sides are known

I tried finding the height of the shaded triangle, which I calculated to be 5. Then I tried solving for the area of the non-shaded triangle, and I got 10 as my final answer. Am I correct?
30 views

### Isometric Drawing Tool: Converting 2D information to 3D

I was drawing with the NCTM Isometric Drawing Tool, and produced the image seen here. I also noticed that it is possible to view the isometric drawing in 2D, and was wondering if/how it is possible to ...
67 views

43 views

### The coordinate of a circle [closed]

Suppose two circles intersect and form three regions A, B, and C. The center of circle A is (2,2) and the center of circle B is (x,y). The three regions formed by the two circles are equal in area. ...
692 views

### How many of these lines lie entirely in the interior of the original cube? [on hold]

A portion of a wooden cube is sawed off at each vertex so that a small equilateral triangle is formed at each corner with vertices on the edges of the cube. The $24$ vertices of the new object are all ...
50 views

### Cutting a Solid Torus with $n$ Planes

Question: How many pieces a solid torus be cut into with three (affine) planar cuts? A google search will quickly reveal that the answer is thirteen, as can be read about here. The picture below ...
84 views

54 views

### Maximal area of equilateral triangle inside rectangle.

If the perimeter of the rectangle is P, what would be the maximal area of the equilateral triangle if: - One of the sides of the triangle coincides with one of the sides of the rectangle - We remove ...
3k views

### Show that a generalized knight can return to its original position only after an even number of moves

Source: German Mathematical Olympiad Problem: On an arbitrarily large chessboard, a generalized knight moves by jumping p squares in one direction and q squares in a perpendicular direction, p, q >...
25 views

### General equation for projection of regular grid onto a line?

I have a regular grid of points in $xy$, say a square grid, and I want to make an orthonogal projection onto a line through the origin, with slope $\tan \alpha$: I would like to derive a mathematical ...
16 views

### Finding Overlap of polygons in 3D space

I'm trying to find the amount of "overlap" between two (or more) polygons in a 3D space. The planes all have vector normals pointing in the same direction, so they are guaranteed to be parallel to ...
113 views

### Exact Differential Equation Geometry

In a variety of contexts, I have noticed hints of a strong connection between exact differential equations and machinery from multivariable calculus. From another question, I have gathered that the ...
20 views

### Decipephering Notation and plugging in values for ellipse formula

Premise I had asked a question on stats exchange about calculating error ellipses for a given scatter plot. I got an answer that seems acceptable, but I'm having trouble implementing it because my ...
consider two oblique oblique basis vectors of unit length $\vec{r_1}, \vec{r_2}$ then any vector $\vec{v} = p\vec{r_1}+q\vec{r_2}$ define the dot product between two vectors a and b as $|b|$ (ie ...