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

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2answers
26 views

Given diagonals, lower base, and height, find the legs and upper base of isosceles trapezoid

Given an the height, base, and diagonals of an isosceles trapezoid, how am I to find the upper base and the legs? I know I can find the area of the triangles made by the diagonals, but how is that ...
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2answers
40 views

Finding an angle between two vectors

I am trying to answer part $d)$ by using my answer to part $c)$. From what I can see, the only possible way to do this is to find the lenght of $AB$ and $OB$, and, using the angle in part $c)$, apply ...
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3answers
39 views

Taking Square Root of Pythagorean Theorem

If one can take of the square roots of both sides of $x^2 = 4^2$ and correctly say that $x = 4$, like here: $$\begin{align*} x^2 &= 4^2 \\ \sqrt{x^2} &= \sqrt{4^2} \\ x &= 4 ...
4
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2answers
86 views

Finding segment in a right triangle.

Here is the picture of the question: $ABC$ is a right triangle. $m(CBA)=90^\circ$. $m(BAD)=2m(DAC)=2\alpha$. $D$ is a midpoint of $[BC]$. $E$ is a point on $[AD]$. ...
2
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1answer
39 views

$20-80-80$ triangle, rhombus with orthocenter, circumcenter

Let $ABC$ triangle such that $\angle A=20^{\circ}$ and $\angle B=\angle C=80^{\circ}$.Let $D,E$ be point on lines $AC,AB$ respectively such that $BD,CE$ are angels bisector of triangle $ABC$.Let $H,O$ ...
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1answer
34 views

Compactification of Lie Group

Is there a way to embed a Lie Group $G$ into a compact lie Group $H$, such that the inclusion is a Lie group homomorphism?
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1answer
31 views

Equation of a plane through line and point

Write the equation of the plane that passes through the line of the intersection of the planes $P_1:x-2y+3z=0$ and $P_2: 2x+z-3=0$ and through the point M(-1,2,6) I take the two planes and I try to ...
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1answer
31 views

question to set relation between angles and arcs in circles

In the following figure how do I prove that $\angle AOC=\frac{\mathrm{arc}AC+\mathrm{arc}BD}{2}$
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2answers
46 views

Calculating segment length on circle

I'm building a physical machine and I'm trying to figure out a geometrical problem. The machine is composed by a cylinder, and the wall of this cylinder is composed by many wooden boards, each of ...
1
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1answer
26 views

Question on circle related to chords and arcs

In the following figure how do I prove that arc AC +arc BD= arc CB +arc AD=semicircle.Given that chords intersect at $90^o$.
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1answer
25 views

Calculation of Rise in Height of water in a Frustum of Right Circular Cone

A volume of frustum of right circular cone is calculated as follows. With known h, R & r of a container with the shape shown below, how to find out the rise in height for each time $7m^3$ of water ...
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2answers
15 views

Mensuration and similarity

Cone P has a volume of 108cm^3 Calculate the volume of 2nd come , Q , whose radius is double that of cone P and its height is one-third that of cone p Here's my working .... $$V_Q=\frac13 \pi (2r)^2 ...
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2answers
29 views

Height of a paralelogramm

I have the coordinates of the 4 vertexes of a parallelogram. If i calculate the length of two opposing sides, how do I get the perpendicular distance between them? Is it just the distance between the ...
1
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2answers
38 views

Circle inscribed in Pythagorean triangle

Given the question (from Burton): "For an arbitrary positive integer $n$, show that there exists a Pythagorean triangle the radius of whose inscribed circle is $n$." My solution is $3n$,$4n$,$5n$ ...
3
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3answers
59 views

How to determine (and explain) the sum of angles without measuring?

Below is a photo of the angles/triangles in which I am working on determining the sum of the angles without measuring. The angles are a,b,c,d,e,f. I understand that angles are formed my ...
2
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3answers
48 views

Find the two other sides in a 15-30-135 triangle

A triangle has angle measures of 15, 30, and 135 degrees. The side opposite the 15 angle is x feet, the side opposite the 30 angle is y feet, and the side opposite the 135 angle is 2 feet. Find x and ...
3
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1answer
57 views

Triangle with same black and white areas

Suppose we have an infinite chessboard with the usual black/white coloring. A triangle $T$ with area $a$ is given with vertices at corners of some cells. Prove that there exists another triangle $T'$ ...
0
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2answers
66 views

Prove that quadrilateral $ADOE$ is cyclic

Let there be a triangle $ABC$ such that $\angle BAC = 60$. Points $D$ and $E$ bisect sides $AB$ and $AC$, respectively. If $O$ is a point in the interior of $ \triangle ABC $ such that $\angle AOB ...
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4answers
1k views

What is the area of the circle?

In the following diagram, $AB = 4$ and $AC = 3$. What is the area of the circle? I can't find any way to solve this.
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1answer
22 views

Clock angle and time passed with minute and hour hand

Let the angle the minute hand covered be $x$ [in degrees] Let the angle the hour hand covered be $y$ [in degrees] I believe that $y = 360 - x$ because of how it is shaped. Hours passed ...
0
votes
1answer
24 views

Right pyramid with a concave polygon as base?

I'm pretty confused about the right definition of right pyramid. I've found both "A right pyramid has its apex directly above the centroid of its base" and "A right pyramid has isosceles triangles as ...
0
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1answer
28 views

Locus of vertex

A variable parabola of latus rectum $l $, touches a fixed equal parabola , the axes of the two curves being parallel . Then locus of the vertex of the moving curve is a parabola , then what is the ...
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2answers
48 views

Number of inscribed triangles in a rectangular hyperbola touching a parabola

How many triangles can be incribed in the rectangular hyperbola $xy= c^2$ whose sides all touch the parabola $y^2 =4ax$. How can we start the question . Please help.
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1answer
27 views

Point of intersection of tangents

If the distance of two points $P$ and $Q$ from the focus of of a parabola $y^2 =4ax$ are $4$ & $9$ then what is the distance of the point of intersection of tangents at $P$ and $Q$ from the ...
11
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1answer
225 views

What is the geometry behind $\frac{\tan 10^\circ}{\tan 20^\circ}=\frac{\tan 30^\circ}{\tan 50^\circ}$?

This identity is solvable by help of trigonometry identities , but I think there is an interesting and simple geometry interpretation behind this identity and I can't find it. I found it when I ...
7
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1answer
44 views

Number of lines formed by sides of polygon

Let $n\geq 3$, and consider an $n$-gon, not necessarily convex. What is the minimum number of distinct lines that are formed by sides of the $n$-gon? When $n=3,4,5$ the answer is $3,4,5$ ...
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0answers
22 views

A book name of geometry [closed]

I'm working know on ACM geometry problems, and I found not good at founding relations and laws. So if you know a book about geometry,please give me the name. thanks
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1answer
30 views

Number of chords having integral length

A point $P$ lies inside a circle centered at $C$ such that $CP=6$. The radius of the circle is $10$. Find the number of chords passing through $P$ which has integral length. Attempt: One solution ...
0
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2answers
33 views

The semiperimeter of an acute triangle is at least the perimeter of its orthic triangle

Let $ABC$ be an acute triangle. If $AD, BE,$ and $CF$ are the altitudes of the triangle $ABC$, prove that $$\text{perimeter of $\triangle{DEF} \leq \text{semiperimeter of $\triangle{ABC.}$}$}$$ ...
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0answers
33 views

Locus of circumcentre

Let $ABC$ be a triangle, and $P$ a variable point on its circumcircle. Suppose $AP$ meets $BC$ at $Q$. What is the locus of the circumcentre of $\triangle BPQ$? Experiments on GeoGebra show that the ...
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0answers
18 views

Axis of a glide reflection

I am currently taking a gap year before starting university and am trying to get a head start by teaching myself some of the course content. As a result I have no one to ask and no solutions to check ...
0
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1answer
15 views

Maximum area of inscribed square

This is follow-on from Minimum area of Inscribed Square If I have square S with perimeter 40, i.e. each side 10, and I have inscribed square T, what is the Maximum area of T? How do I even go about ...
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2answers
28 views

Minimum area of Inscribed Square

GRE study guide asks The perimeter of square S is 40. Square T is inscribed in square S. What is the least possible area of square T? Choices are 45 48 49 50 52 They say answer is 50. How ...
4
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3answers
49 views

Prove the triangle is equilateral

HINTS ONLY please. This is very confusing right off the bat. My guess was that we show the angle $C, M, N$ are all $60^{\text{o}}.$ But I am having difficulty doing as as none of the givens have ...
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0answers
39 views

Prove that either $\angle PQE = 90^{\circ}$ or $\angle PQF = 90^{\circ}$.

Let $ABC$ be a non-isosceles triangle with incenter $I$ whose incircle is tangent to $\overline{BC}$, $\overline{CA}$, $\overline{AB}$ at $D$, $E$, $F$, respectively. Denote by $M$ the midpoint of ...
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0answers
14 views

How to calculate the height of the cuboid tank?

Cuboid shape tank has been filled with 84 liters of water, which makes 70% of the whole tank capacity. What's the height of the tank if its length is 6 decimeters and width 4 decimeters. Can somebody ...
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0answers
5 views

Understanding unit normal curvilinear vectors to the surface of an octant of a sphere

I'm supposed to test divergence theorem on an octant of a sphere for a given vector field. The triple integral part was easy. However, I'm stuck with the double integral part. Now, there are four ...
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0answers
16 views

Look angle as function of segment lengths on two point concurrent circles

Find angles subtended by segments made by intersection of an arbitrary line $$ y = m x + y_1 $$ and set of circles $$ x^2 + y^2 - 2 y k = c^2 $$ passing through, and at, the fixed points $ y= ...
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1answer
15 views

How to determine if an affine transformation would cause reflection?

I have a list of affine transformation matrices and I want to write a code to delete the transformation matrices that applying them on an image would cause reflection. after seeing this image in ...
2
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0answers
41 views

What's so special about involute curves??

An involute curve (specifically, an involute of a circle) is very commonly used to define the shape of the teeth on a gear. Apparently this idea goes back to Euler. Why is this? What special ...
0
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2answers
35 views

Prove that the midpoint of $XY = AB$

Prove that given two point in the plane $A$ and $B$, the midpoint of $AB$ is the same as the midpoint of two points on $AB$, which are $X$ and $Y$ such that $AX = BY$. I have a few ideas for how ...
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2answers
51 views

Excircle and incircle proof

Prove that if the incircle of triangle $ABC$ touches side $BC$ at $D$ and the $A$-excircle touches side $BC$ at $D'$, then the midpoint of $BC$ is the midpoint of $DD'$. This is an interesting ...
3
votes
1answer
23 views

How many combinations of connected midpoints for a regular hexagon?

Board game designer here looking for some help with tile design for a hex-tile based game. any help with my image example or wording to make this question more clear is greatly appreciated. Consider ...
0
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1answer
20 views

Trisecting angle equivalence of constructing a segment

After reading Wikipedia and some previous questions asked in this site, I still don't understand this. Following the Pierre Wantzel. Triple angle formula cos(3theta ) and getting a polynomial p(x). ...
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1answer
48 views

Proof of equal angles in a quadrilateral.

points E and F are given on side BC of a convex quadrilateral ABCD (with E closer than F to B). Suppose angle EAB = angle CDF and angle FAE = angle FDE. Prove that angle CAF = angle EDB.
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1answer
25 views

Geometry problem with rectangular parallelepiped

Given right angled parallelepiped $ABCDA1B1C1D1$, with bases $ABCD$ and $A1B1C1D1$, which are squares with side $1$. if $\angle (B1C;D1A) = 60^\circ$ find the length of the surrounding edge (I'm not ...
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0answers
42 views

How to find a triangle's perimeter only using base and height?

Without measuring the length of the other two sides, is there a way to find the perimeter with one side (Base) and the height of that side?
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0answers
19 views

Metric tensor for just one index

I'm novice in the territory of tensor calculus. I know the utilization of the metric tensor to transform the covariant basis to contravariant one, vice versa: $Z^i = Z^{ij}Z_j$ (Eq. 1) I am going ...
0
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4answers
50 views

Equation of a circle tangent to two lines , given the radius . [closed]

What is the equation of the circle whose center is in the first quadrant and with the radius of $4$ units, given that it is tangent to the $x$-axis and to the line $4x-3y=0$?
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1answer
34 views

Sin and cosine calculations are only calculate on acute angles of triangles?

So sine and cosine calculations are only calculate on the acute non-right angles of triangles. Is that correct? This from math2.org: Definition 1 Given any angle q (0 £ q £ 90°), we can find the ...