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

learn more… | top users | synonyms

3
votes
3answers
58 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
votes
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
votes
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
votes
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 ...
14
votes
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.
1
vote
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
votes
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 ...
1
vote
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.
1
vote
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 ...
10
votes
1answer
215 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 this. Edit : I find ...
7
votes
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 $3,4,5$ ...
1
vote
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
1
vote
1answer
29 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
votes
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 $$\text{perimeter of $\triangle{DEF} \leq \text{semiperimeter of $\triangle{ABC.}$}$}$$ ...
0
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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
votes
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 ...
0
votes
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 ...
0
votes
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 ...
1
vote
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 ...
0
votes
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= ...
0
votes
1answer
14 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
votes
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
votes
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 ...
0
votes
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
votes
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). ...
0
votes
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.
0
votes
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 ...
0
votes
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?
0
votes
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
votes
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 $4x-3y=0$?
0
votes
1answer
33 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 ...
5
votes
0answers
66 views

Optimal escape route out of a half-space in $\mathbb{R}^3$

In $\mathbb{R}^3$, what is the minimum length of a curve starting at the origin whose convex hull contains the unit sphere centered at the origin? I'm looking for an exact answer or bounds. The ...
2
votes
2answers
64 views

What is the maximum number of boxes that can fit in a rectangular container

I'm looking for an algorithm for the following question: What is the maximum number of boxes with sides a,b,c that can fit in a rectangular container with sides $x$,$y$,$z$. For example, the ...
2
votes
2answers
51 views

What is the probability that a unit disk centered at a random point $P$ has exactly two lattice points in its interior?

A point $P$ is chosen at random in the coordinate plane. What is the probability that the unit disk with center $P$ contains exactly two lattice points in its interior? In short I've been trying ...
2
votes
1answer
42 views

Euclidean norm gives length even in $>3$ dimensions?

In $1,2,3$ dimensions I can simply make triangles and see that Euclidean norm gives me the distance between two points (i.e. the length of the vector from one point to the other). In higher ...
0
votes
0answers
13 views

Project 4 cones onto a sphere

I have four cones. The angle of each cones is 140 degree. I need to project it onto a sphere(place it ) such that, the cones cover the maximum area with minimum overlap. I initially thought that ...
1
vote
3answers
25 views

Reflecting coordinates over the line $x = -1$

I know how to reflect a coordinate over the $y$ and $x$ axis, but is there a rule I could use to help me find the reflected point over $x = -1$? This is what I know already: Over the $x$-axis: ...
0
votes
1answer
34 views

Trouble with understanding a solution to an exercise

Given right triangular prism $ABCA_1B_1C_1$, the surrounding edge(not sure if this is the right term in English, but the surrounding edge are $AA_1, BB_1, CC_1$) are equal to $\frac{\sqrt{5}}{5}$ and ...
0
votes
2answers
34 views

Finding the set of points

Let us have a plane, and in it given dots $A$ and $B$, and a given distance $d$. Determine the place of all points $X$ in that plane such that we have $XA^2-XB^2=d^2$. I know that this is elementary, ...
1
vote
1answer
27 views

Catenary Equation of a plane (3D)

A catenary equation models a curve supported by two points, when solely acted on by gravity. The common formula is given as $y= a \cosh(\frac{x}{a})$ where $a$ is a constant regulating the steepness ...
1
vote
2answers
31 views

General form of intersection line between 2D plane and 3D hyperboloid when offset from symmetry axis?

General Background Suppose I have one side of a two-sheet hyperboloid as a general three dimensional shape, where the symmetry axis is along, say, my x-axis ($\hat{\mathbf{x}}$) in my chosen ...
17
votes
4answers
1k views

Why Cohomology Groups?

Why do we need cohomology groups? Homology groups are easier to compute and given two topological spaces, there is an isomorphism in homology groups if and only if there is an isomorphism in ...
1
vote
2answers
60 views

Even function I don't think is even

My professor provided the following function: which he says is even with a period of $2\pi$. I get what he means, it's supposed to be a sawtooth-like function. But is this really even? In a ...
3
votes
1answer
65 views

Smallest Circumcircle of Three Triangles with Equilateral Constraint

What is the minimum diameter of the circumcircle about an equilateral triangle formed by the center points of three congruent equilateral triangles that do not overlap? The diagram is the best ...
1
vote
1answer
54 views

Gaussian curvature expressed by torsion and curvature of its geodesics

Let $S$ be a minimal surface, and let $\gamma$ be a geodesic parametrized by arc length, with curvature $k$ and torsion $\tau$,show that in the points of $\gamma$ $$ -K=k^2+\tau^2, $$ where, $K$ is ...