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
13 views

test if vertical line intersects another

I'm testing if two line segments intersect (given the coordinates of the endpoints). First I calculate the function for each line y = ax + b My question is: ...
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1answer
47 views

Using the Law of Cosines and SSA Congruence to show that a side is uniquely determined

Let $JK = x$, $KL = y$, and $\angle J = \theta.$ Using the values for of $x, y,$ and $\theta$, the third side, $JL$, can be uniquely determined. Using the quadratic polynomial find conditions on ...
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1answer
19 views

Point symmetries around lines in 2D

I am trying to remember how to compute the symmetric point to an $(x=a,y=b)$ point with respect to a line, i.e. $y=mx +b$, without luck. Is there a closed form equation for this type of ...
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2answers
47 views

Is there a name for a point on the circumference of a circle?

Is there an eloquent name for a point located on the circumference of a circle?
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2answers
25 views

Find the scalar projection point with dot product

Take a look at the image bellow: How can I find the point P, without knowing the vector B (or its end point)? Known is: The vector A (start point, end point and its length) Angle theta
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2answers
44 views

Complex number - locus of a point

Question: If argument of $\frac{z - z_1}{z-z_2}$ is $\pi\over4$, find the locus of $z$. $$z_1 = 2 + 3i$$$$z_2 = 6 + 9i$$ Approach: I tried to solve the equation using diagram, basically ...
3
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1answer
80 views

Geometry: How to find cube root, fourth root, fifth root… and so on?

As we know that square root of a number $n$ can be found by using a compass and a straight edge, given the line of length $n$. What I want to know is how to find cube root, fourth root, fifth root or ...
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1answer
33 views

Objectively determining how straight a city's streets are

While discussing the following anecdote on another site Finally (thanks to Iain Macintosh), Daniel Finkelstein in Saturday's Times (pay wall), recalls William Hague's visit to Japan: He ...
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0answers
56 views

Proving that $AB = AC$. [closed]

In a $\triangle ABC$, $D$ is a point on $BC$ such that $AB+BD=AC+CD$.Let the centroids of $\triangle$s $ABD$ and $ACD$, vertices $B$ and $C$ lie on a circle.Prove that $AB = AC$.
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1answer
18 views

How to determine if some line segments are collinear

Let's say I have several Line Segments that are connected to each other and make a Polyline. How can I determine if they are ...
2
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1answer
71 views

Calculating certain functions if only certain buttons on a calculator are permitted

A calculator is broken. The only keys that work are $\sin, \cos, \tan, \cot, \arcsin, \arccos$, and $\arctan$ buttons. The original display is $0$. In this problem, we will prove that ...
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1answer
34 views

Can $a \cos A + b \cos B + c \cos C$ equal $4 \sin A \sin B \sin C$?

This is a follow-up to my previous question about an identity with the sides and angles of a triangle. Can $a \cos A + b \cos B + c \cos C$ equal $4R \sin A \sin B \sin C$? I'm not sure if ...
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5answers
98 views

Find the point on the curve farthest from the line $x-y=0$.

the curve $x^3-y^3=1$ is asymptote to the line $x-y=0$. Find the point on the curve farthest from the line $x-y=0$.can someone please explain it to me what the question is demanding? I cant think it ...
2
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1answer
64 views

Prove $ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $

$ \frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right) $ This is what I have so far: I know that $A + B + C = 180^\circ$, so $C = 180^\circ - (A+B)$. ...
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1answer
30 views

Euclidean problem of geometry

Let the two quadrilaterals ABCD and EFGH been given: Let's take these hypothesis: $AD = EH$ $A\hat{B}D=A\hat{C}D=E\hat{F}H=E\hat{G}H$ $AC=EG$ The triangle ABD is isosceles and equal to the ...
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1answer
53 views

Is Archimedes's method for computing volumes equivalent to Cavalieri's?

Encyclopaedia Britannica is unequivocal: "It turned out that Archimedes had used a method later known as Cavalieri’s principle, which involves slicing solids (whose volumes are to be compared) with a ...
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1answer
13 views

Need help with Cylinder calculations for a game.

I am working on a game where the organisms will be represented by cylinders, and despite my attempts, I'm unable to determine the proper geometric calculations. $v = (\pi)r^2h$ Easy enough! Problem ...
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3answers
36 views

Finding Cartesian coordinates of remaining vertices of triangle, given a vertex and angle from y-axis

I have an isosceles triangle ABC, where the height h and angle at vertex A are known. The Cartesian coordinates of vertex A are also known to be (x,y). If the angle between the y-axis and the line ...
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2answers
43 views

Area of the overlap between a triangle and a square [closed]

$ABC$ is an equilateral triangle, each side has length 4. $M$ is the midpoint of $\overline{BC}$, and $\overline{AM}$ is a diagonal of square $ALMN$. Find the area of the region common to both ...
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2answers
51 views

Optimization: maximum area of a triangle under a parabola

Optimization: maximum area of a triangle in a parabola Inside a curve ($x^2-25$ - Parabola) a triangle is drawn with A as the vertex at the origin and the line joining points B and C lie on the ...
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1answer
51 views

A remarkable area of triangles relationship to be proved

AA' BB' CC' are straight lines drawn from the angular points of a triangle through any point O within the triangle, and cutting the opposite sides at A', B', C'. AP, BQ, CR are cut off from AA', BB', ...
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1answer
24 views

Locus of line segments

If we take a line segment of infinitesimally small length, and draw another line segment of the same length from the endpoint of the first at a particular angle and repeat this infinite number of ...
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4answers
230 views

Is there a name for the curve $t \mapsto (t,t^2,t^3)$?

Is there a name for the curve given by the parametrization $\{(t,t^2,t^3); t\in\mathbb R\}$? Here is a plot from WA. An another plot for $t$ from $0$ to $1$. This curve is an example of a ...
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1answer
19 views

Point coordinates at a fixed distance from a vector

I would like to solve the following generic problem by using vector notation that I will use it to improve my algorithm. I have a vector P1P2 that points P1 and P2 are known. Furthermore, an ...
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0answers
31 views

What kind of statics hyperboleans may have?

Picture that you are a flat-lander and live in the hyperbolic plane. In your land there are flat horses which pull flat carts in arbitrary directions. Total chaos! So there are forces and it is ...
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2answers
32 views

How to prove injectivity of this group homomorphism?

Imagine a cube with endpoints $(\pm 1, \pm 1, \pm 1)$ sitting in $\mathbb R^3$. Color all vertices with even sign. This gives a tetrahedron inside the cube. Let's label the colored vertices ...
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1answer
20 views

Normal vectors of planes intersecting a tetahedron: is my calculation correct?

I need some help with basic linear algebra. The setting is this: Imagine a cube with endpoints $(\pm 1, \pm 1, \pm 1)$ sitting in $\mathbb R^3$. Color all vertices with even sign. This gives a ...
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2answers
33 views

Find the tan A if the triangle is inside the square?

ABCD is a square. The problem asks for me to find $\tan(\angle QAP)$ if I am given the fact that $CP = CQ = \frac{AB}{4}$. This is what I have so far: I drew a line from $Q$ to $P$ to make ...
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1answer
42 views

Check my work on a problem involving Law of Cosines?

The problem is this: Jane walks North for 3 miles, then turns $45^\circ$ to the right. After that, she walks another 4 miles. How many miles will Jane be from her starting point? Give your answer ...
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1answer
51 views

Doubt : Invariance in Geometry

I was working my way through some Proof Problems in Discrete Maths by Rosen, when I came across the following question: What Geometric proposition ( having an invariant property ) does this ...
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2answers
98 views

Proving Concurrency of Midpoints of Segments

The incircle of $\triangle ABC$ is tangent to $AB$, $BC$, and $CA$ at $C'$, $A'$, and $B'$, respectively. Prove that the perpendiculars from the midpoints of $A'B'$, $B'C'$, and $C'A'$ to $AB$, $BC$, ...
2
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1answer
21 views

Does an octahedron have more than $5$ reflectional symmetries?

I counted $5$ planes of reflection for the octahedron: two corresponding to planes orthogonal to two sides and going through one vertex, two corresponding to planes diagonal and one corresponding to ...
2
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1answer
31 views

Diagonal plane of the cube: is it a symmetry?

I'm having a hard time determining whether the plane with normal the middle diagonal of the cube is a symmetry of the cube. I drew pictures but even from the pictures it's really hard to tell. If ...
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votes
4answers
138 views

Why can we use geometric proofs in algebra?

For example, whenever I search for a proof of the Pythagorean theorem, I get a drawing of a geometric proof, yet we use the Pythagorean theorem to algebraically compute distance between points in an ...
3
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1answer
43 views

Find the equation of a line that bisects a set of points

Sorry if this has been asked/answered but I couldn't find anything here or on Google, and for sorry for the poor wording of the title. Anyway, here's my question: Given a set of points in ...
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1answer
80 views

simplify cos 1 degree + cos 3 degree +…+cos 43 degree?

I am currently working on a problem and reduced part of the equations down to $\cos(1^\circ)+\cos(3^\circ)+.....+\cos(39^\circ)+\cos(41^\circ)+\cos(43^\circ)$ How can I calculate this easily using ...
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1answer
27 views

check my work on this problem: given tan(2x), find sin x + cos x?

$\tan 2x = - 24/7$ $90^\circ < x < 180^\circ$. Find the value of $\sin x+\cos x$. What I have so far: $\tan(2x) = -\frac{24}{7} \Rightarrow \frac{2\tan(x)}{1-\tan^{2}x} = -\frac{24}{7}$. ...
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1answer
22 views

Find point in right triangle with given one vector and one point

I am developing a game where the user move a car with his finger. The car is represented as vector (one point and angle of rotation in the screen). When the user start to dragging the car he generate ...
0
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0answers
24 views

Distance from polyhedron to bounding ellipsoid

How can I find the distance from the edges of the polyhedron to the boundary of the ellipsoid? The ellipsoid is parameterized by: $$E = [ x : (x - x_c)^T H (x - x_c) \le m^2 ] $$ And it covers the ...
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2answers
73 views

Prove that $\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}$ for angles of a triangle

Let the three internal angles of a triangle are $a,b,c$. Prove that $$\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}.$$ I'm looking for an elementary, geometric proof. So avoid derivatives and ...
2
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1answer
42 views

Geometry and natural numbers

I can't find the solution to the following problem, any help welcome. One is given a natural number N. One has to find N points on a straight line, and a (N+1)th point which is not on this straight ...
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0answers
48 views

Geometry question: translating a rectangle according to a specific rule

Please take a look at the figure below. I have two line segments: a, which goes from point A to point B, and b, which goes from point B to point C. Each line defines a rectangle, which has width d and ...
2
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1answer
49 views

Difficult elementary triangle problem

I can't find the solution to this problem. One is given the circumscribed circle and the inscribed circle of a triangle, but not the triangle itself. One has to find this triangle, using only a ...
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1answer
29 views

I have a line F(t) = a + bt and a surface, S(u,v). Is there a formula for the intersection between those?

If I have line, F(t) = a + bt (where $a$, $b$ are known 3D vectors), and a surface, S(u,v), as an arbitrary algebraic formula, ...
2
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1answer
29 views

Icosahedron and Cube

We can inscribe a cube in dodecahedron (see this), where $12$ faces of dodecahedron give the 12 edges of the cube. Can we inscribe cube in icosahedron?
3
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1answer
54 views

Nine-point circle equivalent for tetrahedrons?

Nine-point circle for a triangle is defined as the circle that passes through: the midpoint of each side the foot of each altitude the midpoint of the line segment from each vertex to the ...
3
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2answers
79 views

How prove find this value $|AD|+|DF|+|FA|=2$

Question: if $ADB$ and $ACE$ are straight lines with $D,E$ and $B,C$ intersecting at $F$. if $$|AB|=|AC|=1,|AD|+|DE|+|EA|=4$$ show that: $$|AD|+|DF|+|FA|=2$$ I have read this ...
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1answer
20 views

Reflections in polyhedral groups (tetrahedral group)

Consider the symmetry group of a regular tetrahedron (pyramid). I am trying to work out the group of reflections. My observations so far: There are only three reflections. These are the reflections ...
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1answer
63 views

How to translate to a specific point with rotational transformation.

Basically I have two rectangles. ABCD and EFGH EFGH is rotated around it's centre point (X) ABCD has centre point (W) I also know for the sake of this example that EFGH is rotated counter clockwise ...
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0answers
35 views

Cyclic convex quadrilateral property

Let $ABCD$ be a cyclic convex quadrilateral, and let $P$ be the intersection of the diagonals. Show that $\frac{PB}{PD}=\frac{AB}{AD}\cdot \frac{CB}{CD}$. I guess I need to use Ptolemy's ...