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Questions tagged [geometry]

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

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Geometry triangle inside circle

Triangle ABC is inscribed in a circle.D is a point on AC. BD is angle bisector of angle(B). O is the centre of circle then find angle(ADO) if angle(A)=20° and AB=AC.
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Cut circle into percentage proportions

I want to implement a function in javascript but I'm not sure how to construct the math logic behind it. What I need to have happen is position the circle in relation to a vertical line that cut it ...
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Taylor Series and hyperbolic area of a triangle

a) Express cos(A) in terms of cos(α) If area of a hyperbolic triangle is π - α - β - γ, then area of a hyperbolic equilateral triangle is π - 3α. In order to get cos(A) in terms of cos(α), cos(A) = ...
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Show that equilateral triangles are constructible and polygons with 3 * 2^m sides are constructible. [on hold]

Show that equilateral triangles are constructible and polygons with 3 * 2^m sides are constructible. Hint is Bisection of angles is possible. However, I don't know how to use this hint and prove.
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prove the N is midpoint of OH [on hold]

enter image description here please use the secant line theorem wrote above, I think HC'FO is the key to solve this question.
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46 views

Must a subset of $\mathbb{R} ^n$, that has a dimension of $1$, be formed by a union of lines?

Let $S$ be a subset of $\mathbb{R} ^n$ with a box-dimension of $1$. Is $S$ necessarily a union of lines (including "curved" lines and line segments)? If no, then is $S$ at least an infinite set of ...
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1answer
9 views

Variance of x_i chosen from uniformly distributed hypersphere

I'm looking for an expression of the variance of a single component of a point chosen from within a uniformly distributed n-ball with radius r for any n. There are a few proofs showing that ...
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1answer
17 views

Where does the cylinder hit the side of the bucket?

I have two objects. One is a cylinder 305mm in diameter. The other is a bucket which is 330mm at the top and 280mm at the bottom (straight tapered). If I insert the cylinder into the bucket, what ...
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1answer
39 views

How to calculate the integer number of vertices in a 2 dimensional triangle?

Imagine a 2-dimensional right triangle drawn on graph paper (a lattice), with the right corner originating at (0,0). Each unit on graph paper has a width of 1 unit. The lengths of the base and ...
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2answers
238 views

The bible of geometry: Is there a modern treatment of geometries from the most primitive to the most advanced?

About 2000 years ago Euclid wrote a book that contains (almost) all the geometry that was known at his time. Today, in the 21st century, our knowledge of geometry increased drastically: our knowledge ...
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2answers
36 views

In △ABC, AB = 4 cm, BC = 6 cm, and AC = 8 cm. Let D be a point on AC so that BD = AB. Find AD.

In △ABC, AB = 4 cm, BC = 6 cm, and AC = 8 cm. Let D be a point on AC so that BD = AB. Find AD. The correct answer was 11/2 cm and I tried to find out the reason why the answer was such. I first tried ...
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148 views

Inscribed circle in right-angled triangle

In right-angled $\triangle ABC$ with catheti $a = 11\,\text{cm}, b=7\,\text{cm}$ a circle has been inscribed. Find the radius and altitude from $C$ to the hypotenuse. I found that the hypotenuse is $...
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21 views

Distance between two functions must be greater than 1

The functions used for this problem are simplified functions: I have a function $g(x)=x_1^2$ and I have a function $h(x,b)=x+b$ and the Area (let's say in the interval x=[0,5]) between these two ...
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1answer
32 views

Maximum volume cylinder from a sheet of paper

We have an A4 sheet of paper and we have to build a cylinder of maximum volume using this paper by cutting out a rectangle and the two base circles in order to make the construction. My approach: ...
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1answer
10 views

Complement of half spaces covering boundary of a convex body is a polytope

I have the following problem about compact convex sets. Let $K\subset\mathbb{R}^n$ be a compact convex set with nonempty interior. Assume that $A_1,\dots,A_m$ are open half spaces that cover $\...
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1answer
37 views

Geometry question about triangle and a circle

We have triangle $ABC$, and we construct a circle on side AC to become its diameter. This circle contains the middle point of side $BC$ and intersects side $AB$ in point D, in ratio of $AD:DB=1:2$. ...
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2answers
36 views

Divide an acute triangle into $3$ mirror-symmetric shapes

Find three ways to divide any acute triangle into $3$ mirror-symmetric shapes. Two ways, using the circumcenter and incenter, are easy. What is the third way?
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finding circumradius for cyclic quadilateral given side lengths

Suppose a cyclic quadrilateral with sides $a,b,c,d$ is arranged round the circle in that order, say counterclockwise. Is there a formula for the circumradius $R$ in terms of the side lengths? I found ...
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1answer
38 views

Calculate the circle perimeter

I would like please some help with the following Geometry problem: We have a circle and two chords (which are perpendicular to each other). One chord is 3 cm and the other 4 cm. Calculate the circle's ...
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0answers
26 views

Area of Intersection of Two Ellipses

Given two ellipses in space where: a & c are major diameters b & d are minor diameters h & k are x-axis centers j & i are y-axis centers r & s are the rotation of each ellipse ...
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1answer
21 views

Find 1 point from 3 Angles, 2 Points and 1 unit vector

i have the following problem: Given are points P1 and P2. I also have the direction, in the image noted as D, given as a unit vector. Additionally, I have the angles alpha and gamma and I know that ...
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1answer
26 views

Finding integer coordinates for a pentagon, hexagon, heptagon, octagon, and nonagon, etc.

Wondering what the formula is for finding integer coordinates for an arbitrary "regular" polygon. By regular I mean symmetrical polygons like pentagon, hexagon, etc. In particular, I would like to ...
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4answers
51 views

About two inscribed circles in a right triangle

In $\triangle ABC$, $H$ is the foot of prependicular from $A$ to $ {BC}$. $\angle A = \frac{\pi}{2}$. $O_1, O_2$ are the centers of inscribed circles of $\triangle AHB$, $\triangle CHA $ respectively. ...
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1answer
28 views

Number of right angled triangles formed in a cube by joining three of its vertices

We can form a maximum of 4 right angled triangles in a rectangle/square. Due to symmetry any right angled triangle in a cube will also be a part of a rectangle/square formed by its vertices. I can ...
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Surface Area of generalized polar parametrization

Suppose we have a smooth domain $D$ in $\mathbb{R}^n$, which can be parametrized by polar coordinates $(\zeta, r(\zeta))$. Here, $\zeta$ denotes a point on the $(n-1)$ dimensional unit sphere $B$ and $...
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Do multi-focal-point hyperbolae exist?

I recently learned about k-ellipses (see this other question) and I wondered if there was such a thing about k-hyperbolae. I have seen the Wikipedia article on 'Generalized Conics', but I'm not able ...
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1answer
28 views

Demonstration of Menelaus Theorem

I know how to demonstrate menelaus theorem with the similarity of triangles but how do you prove it using ratios? PS: thanks for your time.
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23 views

Finding one point with three distances and three angles in 3D

I am having a bit of a complicated maths problem and am hoping someone might help or have a suggestion. My problem concerns finding the location of a point P in 3D. From this point there are three ...
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1answer
37 views

Could an irregular triangular-base pyramid be constructed with 4 identical irregular triangles? Why or why not?

Suppose we have a regular triangular-base pyramid(A.K.A.: a Tetrahedron). Obviously every single triangular side on a tetrahedron "meets flush" with every single other side. Or, another way of putting ...
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1answer
36 views

Proof that line $HG$ bisects the perimeter of $ABC$

Question. From this picture, $D$ and $E$ are excenters of $ABC$, and $G$ and $H$ are midpoints of $AB$ and $KL$. Prove that $HG$ bisects the perimeter of $ABC$. In other words, prove that $AX=CX+CB$....
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Prove that $BD$ bisects $\angle ABC$

Given that $\triangle ABC$ is an isosceles right triangle with $AC=BC$ and angle $ACB=90°$. $D$ is a point on $AC$ and $E$ is on the extension of $BD$ such that $AE$ is perpendicular to $BE$. If $AE=\...
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Trigonometry equivalent

$\sin (A) = \sin 3x$ is equivalent to $\sin(A) = 3 \sin(x) - 4\sin^3(x)$, then $-\cos 3x$ is $-4\cos^3x + 3\cos$. is that correct? I just want to make sure I'm distributing the negative sign ...
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1answer
27 views

Can four points be a square if they are the same points

My coworkers and I have been doing code golf between ourselves, and this week we're trying to calculate whether four points form a square. Now, we realized we had all missed an edge case, which is ...
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1answer
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Finding a third point of a triangle in 3D

I have 3 vertices in 3D: C, P and W. I know: Points C and P and therefore $\overrightarrow{CP}$ and $\overline{CP}$. A direction vector collinear with $\overrightarrow{CW}$ $\overline{PW}$ I ...
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2answers
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Is there a bounded connected set $X$ such that for all point $b$ there exists $r > 0$ such that $X \setminus O(b, r)$ is disconnect?

I find a question by myself, and I do not know if it is an interesting question. Let $X \subseteq \mathbb{R}^n$ be a bounded connected set. And I define a "bad point" $b \in \mathbb{R}^n$ with ...
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How we can determine the coordinates of rectangle?

Give the center of the rectangle and their sizes h and w respectively and the $\phi$ angle created by h and Oy. How to determine the coordinates of the rectangle. For example: we have the coordinates ...
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1answer
35 views

How can we calculate the volume of a BCC Wigner-Seitz Cell? (Based on a imaginary cube)

Hello. As you can see in the picture, there’s this shape and shape’s surface consists of 6 square and 8 hexagon parts and I would like to know its volume but I don’t know where to start. The only ...
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1answer
42 views

Prove that $Z$ is the midpoint of $OH$

Let $ABC$ be an acute triangle, $O$ the circumcenter, $H$ the orthocenter, $P$ the midpoint of $AO$ and $S$ be the intersection of the perpendicular bisector of $AO$ with $BC$. The ...
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253 views

Quadrilaterals with equal sides

$AC = BD$ $EC = ED$ $AF = FB$ Angle CAF = 70 deg Angle DBF = 60 deg We are looking for angle EFA. I have found through Geogebra that the required angle is 85 deg. Any ideas ...
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29 views

How to determine whether a given point lies inside or outside of a triangle in 3D?

For example if i have triangle defined by following points [A=(15.0, 14, 15.0), B=(15.0, -45, 15.0), C=(-15.0, 14, 15.0)], and consider the point need to be project p=(15,78,0). I want to determine, ...
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19 views

1. Straight lines and triangles

If S is the circumcentre of a triangle ABC and D,E,F are the feet of the altitudes of triangle ABC then prove SB is perpendicular to BF. I have seen solutions to this problem using cyclic ...
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1answer
14 views

Finding Locus of a point under given condition.

Problem Let P be the variable point on a circle C and Q is a fixer point on the outside of C. R is a point in PQ dividing it in the ratio p:q ,where p>0 and q>0 are fixed. Then what will be the locus ...
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1answer
26 views

Straight lines and triangles

AB and CD are two fixed straight lines and a variable straight line cuts them at X and Y respectively .The angular bisectors of angle AXY and angle CXY meet at P.Find the locus of P. My answer is P ...
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1answer
41 views

A locus question [on hold]

A is a give point and P is any point on a given straight line. If AQ=AP and AQ makes a constant angle with AP find the locus of Q. The answer should be a line ,but i do not know how to prove it . ...
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1answer
38 views

Need help with this word problem using hyperbolas and need the final answer in (x,y)

John was in the lead, Ed was 1.5 miles behind and Jeff was 2 miles behind John. Then they heard an explosion. John heard it first and Ed heard it a second later and Jeff heard it 1.5 seconds after ...
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0answers
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Directional Curvature

What is Directional Curvature and how can I achieve it for any function? A common approach with an example would be much appreciated. (Reference: I am reading "The Non-convex Geometry of Low-rank ...
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0answers
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Example of the infinite point density of continua

There is an interesting example that I've seen, possibly attributed to Hilbert. One draws a ray from the center of a circle to every point in the circle, and then one increases the radius of the ...