0
votes
0answers
10 views

Decide coordinates for a vector in a triangle (Image attached)

I have the following triangle. I have to express the line $\overline{AT}$ as a linear combination of $\overline{AC}$ & $\overline{AB}$. A hint was to use the knowledge of $\overline{AT} = ...
0
votes
3answers
29 views

Show that there are exactly two lines through a point p outside the circle that are tangent to the circle C

Let $C$ be a circle of radius $r$ in the plane. Let $p$ be a point in the plane that lies outside of $C$. Show that there are exactly two lines through $p$ that are tangent to $C$. It is one of ...
0
votes
0answers
16 views

Find the intersection of three bisection lines

Let $p_1 = (a_1, b_1), p_2 = (a_2, b_2), p_3 = (a_3, b_3)$ be three, non-colinear points in the plane. For each pair of these points, let $L_{ij}$ denote the line segment from $p_i$ to $p_j$. (a) For ...
1
vote
0answers
23 views

To construct a right triangle given the hypotenuse and sum of two legs [duplicate]

NOTE: I want a hint only. A compass and a straightedge construction:Given a hypotenuse and the sum of lengths of the legs,we need to construct a right triangle. MY TRY: From any ray $BE$, ,let ...
0
votes
1answer
38 views

Volume of the pyramid…

I have such a problem from geometry: Five edges of a regular triangular pyramid have the length of $6$ $dm$, but the sixth- $4$ $dm$. Determine the volume of the pyramid. For me the problem is quiet ...
1
vote
1answer
55 views

Two touching circles inscribed in an angle

There are two touching circles inscribed in a $60^\circ$ angle. The distance between the vertex of angle and the center of smaller circle is $5j$. What is the ratio of the surfaces of two circles?
4
votes
1answer
39 views

Calculate depth using triginometry

I was asked a question like this on an exam today and I'm wondering if I got it right or not. ...
0
votes
0answers
35 views

max and min values on symmetric polytope

Let $-N\leq t \leq N$. Let $A$ be regular $(N-1)$-dimensional simplex with vertices $(t,0, \ldots, 0)\ldots (0, 0,\ldots, t)$ and $B$ be regular $(N-1)$-dimensional simplex with vertices $(t-N+1,1, ...
1
vote
0answers
20 views

Show an immersion is locally one to one using the inverse function theorem

Using the inverse function theorem, show that an immersion is locally one to one. I am really struggling with this homework question can anyone give me a hint?
3
votes
3answers
77 views

Why do we use $cm^2$?

I can't seem to wrap my head around why we should use $cm^2$ for area. According to my textbook we use it for converting units of area but I don't understand how $1cm$ is any different from $1cm^2$. ...
1
vote
1answer
31 views

Find out the $\angle PRQ$

please, help me to solve this.How can I proceed.I just need help. $PQR$ is a triangle. $M$ is a point on $QR$.here,$QM=1/3RM$ , $\angle RPM=30^ \circ$ and $ \angle QPM=20^ \circ$ now,$ \angle PRQ=??$ ...
7
votes
2answers
82 views

Calculating Angles at Vertices

This is quite a tricky question and I can't answer it.
0
votes
3answers
32 views

Sketching coordinates of C when it meets the x and y axis

The question: Curve $C$ has the equation $y=(x-k)^2 (x-k+2)$ Where $k$ is a constant and $k > 2$. Sketch $C$, showing the coordinates of the point where $C$ meets the $x$ and $y$ axes. I am ...
0
votes
2answers
36 views

Geometry: Using Pythagorean Theorem to find lengths of triangle

I am helping someone do their geometry homework, and wanted to make sure I am even doing it correctly. Below is the problem: Here is how I solved it: We know B is the center, so AB=BE. ...
2
votes
3answers
48 views

Geometry: Find the angle x

I am trying to help my little sister do her geometry and seem to have forgotten my basic math skills. Here are a couple she sent me: Any help would be great! Once I get back into the grove I ...
0
votes
0answers
17 views

co-ordinate geometry equations

The center of a circle s is the second quadrant. s touches both the x-axis and the y-axis at the points P and Q respectively s has a radius length of 5 root 2. i. find the equation of s ii. Find the ...
1
vote
2answers
67 views

Finding the measure of an arc on a circle

If someone could work me through how to solve this, that would be great because I am stumped on this one. I know it looks like there is a lot of useless information in the picture, but there are ...
0
votes
1answer
39 views

The relation between the radiuses…

Find $\frac{R}{r}$ where $R$ is the radius of the circumscribed circle of a trapezoid and $r$ is the radius of the inscribed circle of this trapezoid. Thank you!
0
votes
3answers
69 views

Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
1
vote
1answer
57 views

Problem concerning inscribed and circumscribed circles…

Can you please help me solve this really difficult problem: Find R/r where R is the radius of the circumscribed circle of a trapezoid and r is the radius of the inscribed circle of this trapezoid. ...
3
votes
2answers
59 views

What is the most elementary but still correct according to the most rigorous standard proof of the isoperimetric inequality?

Can you write the most elementary proof of the isoperimetric inequality (but still correct according to the most rigorous standard )? $$l^2> 4πA$$
1
vote
1answer
29 views

Convex Sets and extreme supports

Let the set $S$ in $R^n$ consists of the origin $0$ and $n$ lineary independent vectors $T_1, \ldots, T_n$. Show that $C(S)$, the convex hull of of $S$, is the intersection of its extreme supports, ...
0
votes
0answers
26 views

Right triangles, minimum difference of two measured angles

The question reads: If $\angle ABC$ is a right angle and the measure of $\mu(\angle ACB)=r^{\circ}$, what is the minimum difference between $\mu(\angle ACD)$ and $\mu(\angle BAC)$? Which of the three ...
0
votes
1answer
75 views

Prove that the angle is 45

In ΔABC, ∠B is a right angle. D and E are points on segment AC such that AD:DE:EC = 1:2:√3. Then, prove that m∠DBE = 45.
1
vote
1answer
21 views

Given A and B two fixed points on the circle find the locus of the orthocenter of triangle ABC where C is a mobile point on the circle.

First we have to find the locus and then we have to prove double inclusion. Please help me !
0
votes
1answer
37 views

Volume of unit n-dimensional ball, definite integal

As a part of an assignment to calculate the volume of unit n-dimensional ball I got to the following expression, which I believe is true: ...
0
votes
1answer
29 views

Calculate angles of a projection of a tetrahedron

ABCD is a regular tetrahedron. It is projected on a plane in such a way that the projection forms an isosceles triangle ABC (AB = BC ≠ AC) with D lying in the middle of AC (right image): Problem: ...
0
votes
2answers
28 views

Geometry problem without using trigonometry or similar triangles.

I was helping my sophomore friend with her geometry homework and came to problem 13a: My first instinct was to use similar triangles and say $\triangle BKC \sim \triangle CMD$, and from there we ...
0
votes
0answers
24 views

Isoperimetric inequality proof [duplicate]

Can someone give me a neat clear proof (the most simple but rigorous avaiable) of the isoperimetric inequality $L^2> 4πA$?
0
votes
3answers
61 views

In sphere $r \propto \frac{1}{A}$! How is this possible? What's the wrong here?

Surface area $A$ and volume $V$ of a sphere of radius $r$ are \begin{eqnarray} A=4\pi r^2,\\ V=\frac{4}{3} \pi r^3. \end{eqnarray} But then \begin{align} \frac{V}{A} & = \frac{r}{3}\\ ...
0
votes
0answers
15 views

Question about relationship between line and plane in solid geometry?

Given: Cube ABCD.EFGH, with P and Q as midpoints of CG and BF respectively. What is the relationship between: a. EP and plane ABCD b. AC and plane EQPH My attempt: a. Since EP at plane ACGE ...
0
votes
1answer
37 views

How to draw a intersection of two cones graphically

What is the best software we can use to draw the intersections of two cones such as ellipses and hyperbolas mathematically and clearly. Please send some drawaings and software you have used?
0
votes
2answers
32 views

Uniqueness of Point

I want to proove: For points $P \neq Q \in \mathbb{R}$ and $a,b >0$ there is exactly one point $X \in [PQ]$ and exactly one point $X' \notin [PQ]$ with ratio $a:b$. For $a=b$ there is only one ...
1
vote
1answer
44 views

Do I Have To Explicitly Define Points/Lines/Planes? [closed]

The problem I was given is as follows: Assignment: Draw and label the following figures. Two planes that do not intersect. Lines LM and NP on the same plane (coplanar) but do not ...
2
votes
1answer
31 views

Distance of convex combination of pairs of points in $\mathbb{R}^n$

Given 4 points $w,x,y,z \in \mathbb{R}^n$ define for $t\in [0,1]$ $f(t)=d(wt + (1-t)x, yt + (1-t)z)$. Is this function convex? I have found a proof by differentiating twice and calculating a lot but ...
1
vote
2answers
30 views

How to prove that certain points relating to a trapezoid are collinear?

Can you help me to prove that in any trapezoid, which is not a parallelogram, the following points are collinear? The midpoints of its bases. The point of intersection of diagonals. The point of ...
1
vote
1answer
39 views

Finding ratio of external division in a triangle.

Given a $\triangle ABC$ and $P$ dividing $AB$ internally in the ratio $2:3$ $Q$ dividing $AC$ internally in the ratio $1:2$ , with $PQ$ produced and $BC$ produced intersecting in $R$ , to find the ...
3
votes
2answers
85 views

Given the length of two altitudes and one side , find the area of triangle.

Segments $BE$ and $CF$ are the altitudes in $\triangle ABC$. $E$ is on line $AC$ and $F$ is on line $AB$. $BC = 65$, $BE = 60$ and $CF = 56$. Find $A(\triangle ABC)/100$. By the Pythagorean ...
4
votes
3answers
54 views

In $\triangle ABC$, I is the incenter. Area of $\triangle IBC = 28$, area of $\triangle ICA= 30$ and area of $\triangle IAB = 26$. Find $AC^2 − AB^2$

In $\triangle ABC$, I is the incenter. Area of $\triangle IBC = 28$, area of $\triangle ICA = 30$ and area of $\triangle IAB = 26$. Find $AC^2 − AB^2$. Here is a sketch that I drew: From the given ...
1
vote
3answers
155 views

Estimating Eigenvalues and Eigenvectors from an 'eignpicture'

I was given this question at my school but it really does not make sense to me: The unit vectors $x$ in $\mathbb{R}^2$ and their images $Ax$ under the action of a $2x2$ matrix A are drawn ...
0
votes
3answers
86 views

A circle is inscribed in sector of another bigger circle.Given A(circle) find the A(triangle formed by the center and the endpoints of the sector).

Consider sector of circle $MAB$. $∠AMB = 120◦$. A circle $S$ touches side $AM$, side $MB$ and arc $AB$ as shown in the figure. Area of circle $S$ is $75π/(7 + 4√3)$ . Find $4√3$ times the area of ...
0
votes
1answer
28 views

Right -angled triangle perimeter question

A right-angled triangle has an area of 5. The altitude perpendicular to the hypotenuse has a length of 2. Calculate the perimeter of the triangle. I could not get the area of the triangle to be 5 so ...
1
vote
1answer
136 views

Minimum distance to points in plane

Someone told me that the the following problem is elementary. Given three points $a=(-5,0)$, $b=(0,5)$ and $c=(5,0)$ in $\mathbb R^2$ with Euclidean norm: $$\mbox{minimize}\;\; \; f(x)=\|x-a\| + ...
2
votes
1answer
36 views

Determine the angle knowing following…

I have such a problem: In triangle ABC two medians AD and BE intersect in point O. Determine the measure of the angle BAC knowing that triangle AOE is equilateral. I've drawn such a triangle and I ...
2
votes
2answers
49 views

A question related to triangles , areas , ratio of areas of triangles.

I know the title is confusing but that is because of 150-character limit, if anyone of you can improve it , please do. Consider $\triangle ABC.$ Choose a point $D$ on segment $BC$ such that ...
0
votes
2answers
30 views

Geometry problem with 2 circles and a triangle

I tried to solve this problem: But I did not know how to do it so I looked at the answers and I saw E looked convincing because it is the only one that has square powers and D (from the diagram) is ...
0
votes
0answers
25 views

How to find a trajectory of a point

We have given three points in the Euclidean space such as $A=(x_{0},y_{0},z_{0})$ and $B=(x_{1},y_{1},z_{1})$ and $C=(x_{2},y_{2},z_{2})$ and point $D$ moves inside the triangle $\triangle ABC$ such ...
1
vote
2answers
62 views

A problem related to circle , altitude , triangle.

Consider a $\triangle ABC.$ Draw circle $S$ such that it touches side $AB$ at $A$. This circle passes through point $C$ and intersects segment $BC$ at $E.$ If Altitude $AD ...
2
votes
2answers
129 views

In △ABC, median AM = 17, altitude AD = 15 and the circum-radius R = 10. Find BC^2

Question is as per title. Here is a sketch that I made : By Pythagorean theorem , DM is 8. Now how can I calculate BD and MC? I still haven't found a way to utilize the information that the ...
0
votes
1answer
24 views

There is no projective plane of order $10$.

I need to determine if there is a projective plane of order $10$. The Bruck-Ryser theorem tells us that if $n \equiv2, 1 \bmod 4$, and there is a projective plane of order $n$, then $n$ is a sum of ...