For questions about properties and applications of triangles

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3answers
51 views

How many triangles are possible with no side greater than $4$ units?

Consider a triangle having integer sides such that so side is greater than $4$ units.How many such triangles are possible? I could not solve it by trying to use combinatorics. So, how to do it? ...
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21 views

Must the angle bisectors of a triangle all intersect at a common point- proof? [duplicate]

How could you informally write a proof, proving that all the angle bisectors intersect at one common point in a triangle?
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3answers
43 views

Finding all sides of a right triangle from area and a angle

The following calculator can figure out the length of all sides of a right angle triangle using only the area and a angle. How is it doing that..? ...
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2answers
100 views

Triangle Inequality: Upper bound for $\left|\sum_{i=1}^n\frac{x_i}{i}\right|$

Let integer $n>1$ and $x_1,\cdots, x_n\in\mathbb{R}$ such that $|x_1|+\cdots+|x_n|=1$ and $x_1+\cdots+x_n=0$. Prove that \begin{equation} ...
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0answers
20 views

When proving two congruence how do we know which angle/letter comes?

Like lets say this is the picture Like how would i know which angle comes first for example After proving which angle congruent to which, how do i write. Triangle VTU =/congruent to CTB or Triangle ...
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1answer
13 views

Given the following transformation rule and the coordinates of the image, find the coordinates of the Pre-Image

So what I got was that they want me to reverse everything I think not sure.. R_x-axis:Triangle ABC = T(x,y) to (x, y-3) The points given for the image of the following are $$A''(2,5) B''(-4,2) ...
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22 views

How to find a valume of a prism when when we are not given the height?

So I was learning how to find the surface area and volume. I came across few youtube tutorials which were simple. And in my book I found much harder problem. It asks me to find the volume of the prism ...
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1answer
40 views

Show that in this right angled triangle,$x^0+y^0=z^0$…

$ABC$ is a right angled triangle at $B$.On side $AB$ points $E$ and $F$ are taken such that $AE=EF=FB=BC$.Let, $\angle CAE=x^0$,$\angle CEF=y^0$ and $\angle CFB=z^0$. Now,prove that ...
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1answer
22 views

Finding $\theta$ in this geometric construction

I am working on a laser wavefront analyser and I need to calculate the angle of tilt of a diffuser in my setup which I illustrated below : The screen is shown in green an the angle I want to ...
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2answers
33 views

Find the leg of an altitude in a triangle

The vertices of $ABC$ are $A(8,5)$, $B(0,1)$ and $C(9, -2)$. Find the point where the altitude from $A$ intersects $BC$. Progress: I have found the equation of the altitude from A to BC, and that ...
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1answer
22 views

Calculate the distance from Plickford to Murbell

Attached is my question. Please provide an explanation for how I could calculate the distance from Plickford to Murbell.
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3answers
80 views

How do you find the area of the shaded (gray) region of the square not getting overlapped by the circle or triangle [closed]

How do you find the area of the gray region in the problem. Pretty much the isosceles triangle is 2" tall and 2" wide at the bottom. The circle has a radius of 1" The square is 2" tall and 2" wide. ...
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1answer
29 views

Points of squares from triangle sides on circle

Taking a course on geometry, got this problem in my problem set. Suppose we have a triangle ABC and we take squares $BCP_1P_2$ and $ACP_3P_4$ such that $P_1, P_2, P_3, P_4$ are all on the same ...
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2answers
70 views

Show that no Pythagorean triangle can have its area equal its hypotenuse

One problem we were given in our number theory course was to show that no Pythagorean triangle can have its area equal its hypotenuse. Here is my attempt: Let the sides of the triangle be given by: ...
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2answers
57 views

trigonometry: why this is not a triangle?

As the photo shown, angle x doesn't nesscarry to be 90 degree.But when y=0, x=90 degree. So here is my problem, what happen if I increase y, does x becomes larger (greater than 90) or less than 90 ...
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1answer
43 views

Prove that for every two triangles, there exists a line that halves the areas of the triangles simultaneously.

The problem may sound somewhat funny, cause I haven't got it from a good source. Anyways, I think I somehow get what it wants and here's my way of looking at the solution: Two triangles of arbitrary ...
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1answer
18 views

Find the expected area of a randomly chosen triangle.

The set of numbers $(x,y)$ are positive natural numbers such that $x+y=n$. 2 points are chosen from this set. What is the expected area of the triangle formed by the origin and the two points?
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1answer
63 views

question on right angle triangle

Let ABC and DBC be two equilateral triangle on the same base BC,a point P is taken on the circle with centre D,radius BD. Show that PA,PB,PC are the sides of a right triangle.
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1answer
68 views

Area of the triangle determined by the line $x+y=3$ and the bisector of angle between the lines $x^2-y^2+2y=1$

What is the area of the triangle formed by the lines $x+y=3$ and angle bisectors of pair of straight lines $x^2-y^2+2y=1$ . I found the intersection point of these equations $(1,2)$ but not getting ...
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1answer
34 views

Area of a trapezium

Given the following trapezium: Where area of the triangle BCP is equal to 12 and $|DC|=7$, $|AB|=28$ Calculate the area of a trapezium. How to do that? as it seems that there are no sufficient data ...
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1answer
24 views

Triangle property intersection proof with vectors

Given that: $\vec{OA}=a$ $\vec{OB}=b$ $\vec{OC}=c$ And that point R and F are the mid-points of $\vec{AC}$ and $\vec{BC}$ respectively, show that $\vec{OG}=\frac{1}{3}(a+b+c)$. I am really ...
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1answer
62 views

About isosceles triangles

Let $ABC$ be an acute-angled triangle in which $\hat{ABC}$ is the largest angle. Let $O$ be its circumcenter. The perpendicular bisectors of $BC$ and $AB$ meet $AC$ at $X$ and $Y$ respectively. The ...
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1answer
19 views

Finding all coordinates of a triangle on a graph using three vertices.

I'm working on computer program and I need to know how to find all of the coordinates (x, y) of a triangle given on a graph using the three vertices. The Triangle may be any type including right, ...
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2answers
41 views

How do I find a third side of a triangle with two sides and a bisecting line segment?

I am using a laser range finder to calculate the height of a second story wall. I have a fixed point and three separate lengths hitting the top, the bottom, and an indeterminate point on the wall. ...
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1answer
42 views

Rational distance from a regular polygon.

Consider a regular n-gon with side length $A$. Let $p$ be a point in the polygon. Let the distances from $p$ to the corners of the n-gon be $x_1,x_2,...,x_n$ Are there solutions with ...
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0answers
29 views

Area of triangle given 3 equations of sides, without finding points of intersection [duplicate]

Can anyone tell me a way to calculate area of a triangle given the equations of its 3 sides without sketching or finding the points of intersection?
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2answers
45 views

If point O lies inside triangle ABC then $\left\|\frac{\vec{OA}}{|OA|}+\frac{\vec{OB}}{|OB|}+\frac{\vec{OC}}{|OC|}\right\| \leq 1$

Given triangle ABC and a point O lies inside the triangle. Prove that $$\left\|\frac{\vec{OA}}{|OA|}+\frac{\vec{OB}}{|OB|}+\frac{\vec{OC}}{|OC|}\right\| \leq 1$$ I am undergraduate student and I ...
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2answers
90 views

Concurrency of lines formed by pair of circles joining pairwise.

How can i prove that lines $AD,EB,CF$ are concurrent ? My attempt Considering $\Delta ACB$ I've got the condition that $\cfrac {CP \cdot BQ \cdot AR}{PB \cdot AQ \cdot RC}=1 $ ,but I don't see how ...
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27 views

Prove that $\frac{t}{3} < Rr$.

The task is: An optional triangles area should be $t$ and its circumscribed circle radius $=~R$ and its inscribed circles radius $=~r$. Prove that $\frac{t}{3} < Rr$. I was trying to solve it by ...
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0answers
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Cutting triangles into two pieces

The task is this: We cut an obtuse triangle into two pieces by cutting it from its largest angle vertex into two isosceles triangles. How large are its angles (of the original triangle) if we could ...
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1answer
41 views

How many triangles can be created from a set of points such that no lines cross

I'm pretty sure this is a graph theory problem : I'm trying to figure out a formula for the number of triangles that can be created from n distinct points in the (Euclidean) Plane such that no 2 ...
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4answers
85 views

Find the area of a triangle given the coordinates of its vertices

The following is a GRE practice test question which I don't understand. According to this general argument, it appears that any arbitrary point $(x, 7)$ would have "height" equal to 10 but then if ...
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1answer
51 views

Find an angle in a triangle with cevians

Given triangle ABC such that angles B and C both measure 70 degrees, points E and F lie on sides AB and AC, respectively, such that angle ABF measures 30 degrees and angle ACE measures 50 degrees. ...
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0answers
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Solving the Height of a Triangle at a Point

I just solved a right triangle with all the angles and measurements: I want to find the height of the triangle at 2ft: How would I find this and what would the height be?
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1answer
75 views

Are lines which pass respectively through vertices $A,B,C$ and incenter, circumcenter and orthocenter of $\Delta ABC$ concurrent?

Prove that the lines through $A$ and the incenter of $\Delta ABC $, through $B$ and the circumcenter of $\Delta ABC$, and through $C$ and the orthocenter of $\Delta ABC $ are concurrent if and only if ...
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3answers
40 views

Equilateral triangle and another triangle with same perimeter. Which has larger area?

There is an equilateral triangle with sides $a$ and another triangle with sides $p,q,r$, both having the same perimeter $S$. How can we mathematically show which of them has a larger area?
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2answers
51 views

Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$

Constructing triangle $\triangle ABC$ given median $AM$ and angles $\angle BAM, \angle CAM$ I start with the median $AM$. Since $\angle BAM, \angle CAM$ are known I can construct them. So I have ...
4
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1answer
37 views

How to prove the concurrency of these lines?

Suppose that in $\Delta ABC$ we take a point $D$ on $BC$ such that the incircles of $\Delta ACD$ and $\Delta ABD$ are tangent at a point.Now suppose that points $H$ and $I$ are defined on segments ...
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1answer
43 views

Finding Angles Given Two Triangles with Equal Perimeters

The two right triangles shown below have equal perimeters The hypotenuse of the orange triangle is one leg of the green triangle stacked on top of it. If the smallest angle of the orange triangle is ...
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1answer
25 views

Given that the triangle alongside is equilateral, find a and b.

Given that the triangle alongside is equilateral, find a and b. First side equals to (b+2) The second side equals to (a+4) The third side equals to (4a-b) Thank you for the help :)
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1answer
28 views

Whats the terminology for defining a point on a triangle?

There are 2 common methods of representing a point on a 2d/3d triangle. 2 numbers (often called "UV coordinates" in 3D graphics):Where 2 edges of the triangle are axes, which the point is translated ...
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2answers
60 views

How to prove that these lines are concurrent?

Point $D$ is chosen on side $BC$ of $\Delta ABC$ such that the incircles of $\Delta ACD$ and $\Delta ABD $ are tangent at $G$. Let line $l$ be the angle bisector of $\angle ABC$ ,line $m$ be the ...
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1answer
71 views

Triangle Inequality in Complex Analysis

Consider an equilateral triangle T. Suppose that f is analytic inside of T and satisfies that |f(z)| ≤ 8 on one side of the triangle, while |f(z)| ≤ 1 on the other two sides. Prove that |f(c)| ≤ 2, ...
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1answer
16 views

How to find the length of paralell line crossing the centroid of triangle

Given: Triangle ABC AB = 6 MN(M belongs to AC, N belongs to BC) is parallel to AB and it's crossing the centroid of the triangle. Find the length of MN I'm ...
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1answer
30 views

How to determine the sides of this triangle

Given: ABC is right triangle CH - height of hypotenuse ABC is similar to the triangle ACH and CBH The area of ABC = 30 AB = 13 Find: ...
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1answer
23 views

How to obtain 3D coordinates of the point by the length of the vector?

How can I obtain R3 position of a point? For example, I've got two points linked by a vector: p1 = (-4000;250;-5000) p2 = (428;776;-300) |v| = 6926.32 I'd like to find a point which lies on the line ...
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3answers
83 views

Finding angle in a given triangle.

In the picture above: $\overset{\Delta}{ACD}$ is a triangle. $B$ is a point on $[CD]$. $m(\widehat{ABC})=140^\circ$ $|AB|=|BC|$. $|AC|=|BD|$. What is ...
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2answers
69 views

What proportion of triangles are acute?

An acute triangle is one in which all angles are acute, ie all angles $<90^\circ$. What proportion of triangles are acute? I have two attempted answers which are different; I suspect this may be ...
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1answer
94 views

I have X feet of rope. What should maximum size of triangle be?

Since we're approaching the Christmas season, I'm calculating how many feet of lights I need for a few decorations. Let's say I have X feet of lights, is it possible to calculate the height/width of ...
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1answer
28 views

Finding areas of triangles

Given that: The triangles ABC and A1B1C1 are similar. The ratio of two sides is 6:9 The sum of their areas is equal 52 (Sabc + Sa1b1c1 = 52) Find the two areas ...