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

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1answer
14 views

How to find the radius of this middle circle arranged as shown.

There is this maths competition geometry problem and my approach. And this is my initial approach. From the picture, the shaded circle looks slightly bigger. What we are looking for is the $x$ ...
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2answers
30 views

A geometric question about drawing lines in a plane

Suppose we were to draw lines in a plane such that no two of them are parallel and no three or more meet at one point, so in other words there is only double intersections. If we drew $x$ lines how ...
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1answer
7 views

Angles of which locations of points corresponding to distances intersect each other

Original problem : In the $XY$ plane,Let $P_1$ and $P_2$ two points with coordinates $(-1,0)$ and $(1,0)$. $C_1$ is the locus of points whose sum of distances to $P_1$ and $P_2$ is $4$. $C_2$ is the ...
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1answer
22 views

Example of a proof using the axiom of commensurability

I'm teaching our intro to proofs course (well, one of them) and one of the classic illustrations of an overturned "axiom" is the Greek axiom of commensurability, which stated in geometric terms the ...
2
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2answers
26 views

Right-Angled Isosceles Triangle covering puzzle

Consider a RAIT (right-angled isosceles triangle), from which we remove a RAIT smaller than half its area by a cut perpendicular to the hypotenuse, like this: How many RAITs are required to cover ...
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1answer
20 views

Algorithm for solving line line intersection in 3d

I am trying to find an algorithm that a computer can execute that finds the intersection point between two lines each defined by a point on the line and a direction vector. Does anyone know of one? It ...
2
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1answer
11 views

inflection point of cubic bezier with restrictions

Say you have this type of cubic Bézier curve: The 4 control points A,B,C,D have restrictions: A & B have the same Y-axis coordinate C & D have the same Y-axis coordinate B & C have ...
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1answer
29 views

Why does $n$ have to be a perfect square for me to construct an equilateral triangle out of equal smaller ones?

If I have $n$ unit squares and want to build a bigger square out of the ones I already have, it is obvious that $n$ itself has to be a perfect square. But after doing some elementary math it turned ...
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0answers
13 views

Alternate proof of Girard's theorem

I am looking for an alternate proof of Girard's theorem. The standard proof relies too much on visualization spherical triangles on the sphere. Is there a more algebraic proof?
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2answers
44 views

How to determine whether three ellipses have at least one common intersection point or not?

How to establish a criterion described in equation so that it is easy to determine whether three ellipses have common intersection area (point) or not? Update
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6answers
2k views

Can one deduce whether a given quantity is possible as the area of a triangle when supplied with the length of two of its sides?

Recently I have found a question like following: In triangle ABC, AB=AC=2. Which of the following could be the area of triangle ABC? Indicate all possible areas: [A] 0.5 [B] 1.0 [C] 1.5 [D] ...
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1answer
27 views

Finding equations for plane figures using complex coordinates

I have to find conditions defining the following plane figures: Where: $a=3$ and $b=7$ I know that circumference form is: $$\left |z-z_0 \right | =b$$ So, for c. with center $(3,3)$ and radius ...
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2answers
33 views

No of triangles in a square which contains all the m points?

Given a square $A(0,0)$, $B(0,n)$, $C(n,n)$ and $D(0,n)$ in $X­Y$ plane and a set of $m$ points. The $m$ points strictly lie inside the square $ABCD$. It is clear that there are $4n$ integer points on ...
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1answer
14 views

Avg # of Rectangle Intersections in 2D Field

So imagine I have a large 2D field. Thousands of small rectangles overlay the field. The field is much larger than the rectangles. The rectangles are placed randomly in the field such that they may or ...
1
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1answer
29 views

At the instant of release of an object from rest. Is the only force that can act its weight? [on hold]

This is the third question from a mechanics exam past paper: I can do parts i) and ii) but for iii) in finding the angular acceleration, i used $C=I\alpha$, where $C$ is the applied couple or ...
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0answers
8 views

Extract equations of dependency between two projected views

Regarding question Finding equation of an ellipsoid, the answer says that we have the following equation between projections on XY & XZ plane: $$\frac{Z_3^2}{Z_2} - Z_1 = \frac{Y_2^2}{Y_3} - Y_1$$ ...
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2answers
30 views

Quadrilateral angles when inscribed inside square

"If we can draw a quadrilateral inscribed in a circle, its opposite sides must sum to 180∘." Why is this?
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2answers
40 views

Calculate the unknown coordinates of a point $B (x_2,y_2)$ on a line with given distance from a known point $A(x_1,y_1)$

I have a line which represents a cross section. I have the coordinates of on its starting point. I need the coordinates of the end point of that cross section line. The distance between these two ...
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2answers
38 views

Why does resolving forces in one direction give a completely different answer to resolving the opposite way?

I can solve parts i), ii) and am able to show that $R=0$ for part iii). In this question $g$ is the acceleration of free fall taken to be $9.8$ Using Newtons 2nd law [$F=ma$] for the last part I ...
2
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1answer
16 views

Does simply-connected imply measureable?

The famous examples of non-connected sets involve a sophisticated selections of points from a ball (or another object). This raises the following question: if a certain object in a Euclidean space is ...
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1answer
20 views

Prove that line segments are parallel.

Prove using slope of lines that line segment joining the midpoint of $\overline { AB}$ and $\overline{AC}$ in $\Delta ABC$ is parallel to $\overline {BC.}$ Need to prove using slope of lines means I ...
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3answers
47 views

$T^2\times S^n$ is parallelizable

This is taken from a UCLA Geometry/Topology qualifying exam. How would one prove that $T^2\times S^n$ is parallelizable for all $n\geq 1$? Is there a way to find $n+2$ linearly independent vector ...
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1answer
44 views

Geometry question pertaining to a plane going through the skeleton of a cube

My question is as follows: a plane that has taken the shape of a pentagon is intersecting the skeleton of the cube. Or I guess we could think of it as a cross section. Points $M$ and $N$ were used ...
2
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0answers
37 views

Find circles that completely cover a polygon minimizing the amount of space covered outside the polygon

I have an arbitrary polygon that I need to roughly represent using circles. Any point inside the polygon must lie inside a circle. There will be points outside the polygon that will fall under a ...
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0answers
30 views

Strange question about magnetic dipole in a plane at infinite distance

Please allow me to ask something rather unusual and perhaps completely naive. Suppose I have an electric current in a circular loop in a plane. Consider it just in a mathematical sense. The loop has ...
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2answers
34 views

Prove triangles formed by two midpoints and an altitude are congruent

Triangle ABC has altitude BH. M is the midpoint of AB, and N is the midpoint of CB. Prove triangle MBN is congruent to triangle MHN. Can we say that MN bisects BH? If so, why? If MN bisects BH (at ...
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0answers
39 views

Arc length for a function $f:\mathbb{R}^2 \to \mathbb{R}^2.$

Assume $f:\mathbb{R}^2 \to \mathbb{R}^2$ is $C^1.$ Is there a formula for the length of the subset of $\mathbb{R}^2$ given by $$ \{f(x,y) \in \mathbb{R}^2:a_1\leq x\leq b_1, a_2\leq y \leq b_2\} ? $$ ...
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1answer
10 views

calculate the coordinates of the intersection between a bisector and a sector

I have a sector and would like a formula that gives the intersection between the bisector and the arc. here's a graph of the situation: the point B is the center of a circle of radius AB and the BD ...
1
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1answer
39 views

Prove triangle made from two altitudes and midpoint is isosceles

In triangle ABC, AH and BK are altitudes. M is the midpoint of AB. Prove that triangle MHK is isosceles. All I can see is that the angles formed where the altitudes intersect are equal, and since ...
0
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1answer
50 views

How to make the Symmetric Distance a metric?

I am trying to construct a family $S$ of measureable subsets or $R^2$, on which the symmetric difference, defined as: $SD(A,B) = Area(A\setminus B \cup B \setminus A)$, is a metric, i.e., different ...
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1answer
93 views

Is a Squircle a circle?

Someone has claimed that a squircle is a circle but as far as I understand squircle is just something between a circle and a square. What is correct?
1
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1answer
39 views

Fundamental questions on fundamental classes

Background: Hatcher's algebraic topology book. I would like to know whether, being given a closed orientable $n$-manifold with $\mathbf{Z}$-fundamental class $[M]\in H_n(M)$, there is a relation ...
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2answers
66 views

In any triangle ABC, the expression (a + b + c) (a + b - c) (b + c - a) (c + a - b)$ is equal to

In any triangle ABC, give an equivalence to the expression $$(a + b + c) (a + b - c) (b + c - a) (c + a - b)$$ Can somebody help me? Note that ...
2
votes
2answers
51 views

Hilbert's construction of multiplication of two numbers

I am now reading Hilbert's "Foundations of Geometry", section 15, where he describes there a geometric way to construct, given two segments of length $a$ and $b$, a segment of length $ab$ (in short: ...
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0answers
34 views

Using two chords and an angle to find center and radius of a circle

Hello, I am trying to solve the problem below. Is it possible to solve for the Center and Radius of the circle given the information provided, or is there something missing? I know how it's simple ...
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0answers
14 views

Catenary and parabola minimum comparison

Do the catenary and a parabola that approximates the catenary, have the same minimum (maximum sag)? IF plotted, it looks to me they do, and that they only difer somewhere on the "slope". (sorry for ...
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0answers
9 views

Add a rotation to latitude/longitude -> screen coords?

I found an algorithm which allows to convert latitude/longitude to (x, y) of a screen. The problem is a picture on a map is rotated. If it is not rotated then I use the following calculations: ...
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1answer
39 views

What does “generate freely” mean?

Given a number field $K$ (i.e. $\mathbb Q\le\ K\le\mathbb C$, $[K:\mathbb Q]=n$), the relative number ring is $R=\mathbb A\cap K$, where $\mathbb A$ is the ring of the algebraic integers in $\mathbb ...
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0answers
17 views

3-Point Shoot using Quadratic Equation [on hold]

This is my assignment. The question is "In what part of the three-point line can a player do best the three-point shoot to gain 3 point but using quadratic equation." There are no data given but we ...
5
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1answer
76 views

Is any submanifold of $\mathbb{R}^{n}$ the zero set of some polynomial?

For a given circle we have a corresponding equation to generate it. For a given ellipsoid we also can write a corresponding equation for it. In general, can we write for any given manifold an equation ...
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0answers
16 views

Geometry (finding the coordinates) [on hold]

Find the coordinate of B for each ofthe following. AB=15 and the Coordinate of A is 7
4
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2answers
56 views

Expressing the area as a function :)

Express the area A of an equilateral triangle as a function of the height of the triangle. Thanks :) I am not sure where to even start on how to answer this problem.
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2answers
16 views

Ellipse as projection of a disk - function describing ellipse diameter with disk rotation?

Say I have got a disk of radius $r$ and a plane $p$ in $3D$ space. The disk is "aligned" to $p$ and lies at an arbitrary distance, so that its orthogonal projection on $p$ is an identical disk of ...
2
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1answer
27 views

Find area of a magnified polygon

I have a highly irregular polygon made out of polylines for which I KNOW the area A1, the perimeter P1. I can offset this polygon (magnify it) by offsetting every site by a given distance d. Is there ...
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2answers
20 views

Medians of a triangle and similar triangle properties

Prove using similar triangle properties that "any two medians of a triangle divide each other in the ratio $2:1$. I do not know which criteria of similar triangle must be used
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0answers
27 views

Not-pairwise disjoint sets in $\mathbb R^n$

Consider a fundamental parallelotope $F$ for a $n$-dimensional lattice $\Lambda$ and consider a convex, measurable, centrally symmetric subset of $\mathbb R^{n}$, which we call $E$, such that ...
2
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5answers
120 views

Is there a pure geometric solution to this problem?

$ABCD$ is a square and there is a point $E$ such that $\angle EAB = 15^{\circ}$ and $\angle EBA = 15^{\circ}$. Show that $\triangle EDC$ is an equilateral triangle. Now there is a proof by ...
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1answer
30 views

Main theorem of Pythagorean plane

The theorem states: Any Pythagorean plane is isomorphic to the Cartesian plane $F^2$ over its field $F$ of segments. Can anyone give me a reference for this theorem?
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1answer
17 views

Proof with triangle with similar triangles formed by medians

Use similar triangles to prove that $|AS|:|SD|=|BS|:|SE|=2:1$. My try: $|AE|=|EC|$, $|BD|=|DC|$ $\Rightarrow \triangle ABC \sim \triangle EDC$ $\Rightarrow \angle A=\angle E \Rightarrow AB ...
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1answer
14 views

Parallel property of semi-Elliptic plane

Semi-Euclidean plane means that the sum of angles of triangle is two right anles,which correspnds to the Euclidean parallel property. While semi-elliptic plane means the sum is larger than two right ...