For questions about properties and applications of triangles

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3answers
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Trigonometric problem in triangles.

I need your help. I'm studying physics, but I have a trigonometric problem. I attached a figure where depicts the angles and the unknown $x$. The idea that I want to understand is how to express $x$ ...
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4answers
444 views

What is so special about triangles?!

Take any random triangle. If we draw internal-angle-bisectors of all its angles, they intersect at the same point. If we draw the perpendicular bisectors of each side (although they aren't ...
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1answer
738 views
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1answer
123 views

Triangles within square

Points E and F lie on the sides BC and CD of rectangle ABCD, the AEF is an equilateral triangle. point M is the midpoint of the AF. Prove that the triangle BCM is equilateral.
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1answer
62 views

(Non-)Uniqueness of a given triangle

Let's assume that some triangle is described by its two sides and an angle (which is not between the given two sides however). Basically for a triangle above only characteristics $c,\ b,\ C$ are ...
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2answers
234 views

Rectangle divided into three triangles with two lines. One angle is given, what are all the others?

Let's suppose I have a rectangle divided into three triangles in the following way. No lengths of either the rectangle or triangles are known, only one angle is known. I would like to know how to ...
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1answer
109 views

Correct my Pre-Calculus work please?

PPlease explain how to do these problems. I got my test back and I'm trying to see what I did wrong so I can do better on the next test. Given $g(x)=\sqrt{x+5}$ find $g^{-1}$ $x= \sqrt{y+5}$, I ...
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2answers
114 views

When does the triangle have the smallest area?

The following triangle has an area $S$, and the sides $AO$ and $BO$ have the length $a$ and $b$, respectively. There is a fixed point $X$ at $(x,y)$. A point $C$ is put on the line segment $OA$, and ...
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1answer
92 views

Good websites/books for geometry exercises?

I'm looking for exercises similar to those seen on putnam exams or olympiad exams, such as finding the area of polygons inscribed other polygons, finding certain angles, etc.
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1answer
20 views

Simple Triangle Completion

How do you find the missing point of a triangle, given: 2/3 of the points, two slopes, and one angle of direction. Here's the problem. There are two points: Point B (1,1), showing an arrow going ...
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2answers
466 views

Use the law of cosines to derive the triangle inequality

I am given the vectors: and show that they span the triangle with sides $a,b,c$ with $c=||u-v||$ and determine for which $\gamma∈[0, \pi]$ we have equality. Any help is appreciated.
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4answers
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finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in x-y plane? One approach is to find the length of each side from the coordinates given ...
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2answers
773 views

Find an angle of an isosceles triangle

$\triangle ABC$ is an isosceles triangle such that $AB=AC$ and $\angle BAC$=$20^\circ$. And a point D is on $\overline{AC}$ so that AD=BC, , How to find $\angle{DBC}$? I could not get how to use ...
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1answer
103 views

Getting an angle

I have a unit circle, and two angles: $\alpha=\angle{JON}\in[0,\pi]$ and $\beta=\angle{IOM}\in[0,\frac{\pi}{2}]$. Using angles, we can get points $N$, $M$ as on the image. Then, dropping a ...
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1answer
69 views

About the area of integer-edge-length triangles

Let $a,b,c$ be three edge lengths of a triangle whose area is $S$. Then, here is my question. Question : Supposing that $a,b,c$ are natural numbers, then does there exists $(a,b,c)$ such that ...
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2answers
92 views

Two objects travel on a 2 dimensional grid. How can i find the angle that must be taken in order for the interception time to be the smallest [closed]

An object (a) travels on a linear path at constant speed. A second object (b) must intercept object a in the shortest amount of time possible. Object b is also at a set speed and can travel in any ...
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1answer
116 views

Triangle problem (only know 1 side and y-axis coordinates of 2 points)

I have a triangle problem that I am in desperate need of help with with. Here is what I know ... the triangle is mapped on a graph where I only know y-axis coordinates for 2 points, the length of one ...
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2answers
87 views

Generating Pythagorean Triples S.T. $b = a+1$

I am looking for a method to generate Pythagorean Triples $(a,b,c)$. There are many methods listed on Wikipedia but I have a unique constraint that I can't seem to integrate into any of the listed ...
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1answer
68 views

non right angle triangle - solve for B when b, A and a are known

I'm trying to work out the two possible values for B when A, a and b are all known. I'm certain its possible but I'm not sure how to start or what theories to look for to solve the question as my math ...
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2answers
655 views

How to find sum of 3 perpendiculars of a triangle?

Q. ABC is an equilateral triangle with side 10cm and P is a point inside the triangle, at a distance of 2cm from AB. If PD, PE and PF are perpendiculars to the three sides, find sum PD+ PF+PE. ...
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1answer
137 views

Lines $ MF, DE, QR$ in a triangle intersect at one point

In a triangle ABC, a circle is inscribed with center in $I$. The inscribed circle touches sides $BC,CA,AB$ in $D,E,F$ respectively. Join the point $C$ and $F$, $B$ and $E$. Let $Q$ and $R$ be the ...
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4answers
278 views

How do you find the area of a triangle in a 3D graph?

How do you find the area of a triangle in a 3 dimensional graph? Is it any different than a regular 2d graph? How would you solve it, if these were your three points? A(1,-4,-2), B(3,-3,-3), ...
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4answers
142 views

Is there an integer that $\sqrt{3}$ can be multiplied by that will produce a whole integer?

The question came up while messing around with graph paper. I wanted to make an isosceles triangle where the length of one side and it's hight were both integers. The closest I could get was a base ...
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2answers
1k views

How to find the type of triangle when given the ratio of it's sides?

Q.The sides of a triangle are in ratio 4 : 6 : 7, then the triangle is: (A) acute angled (B) obtuse angled (C) right angled (D) impossible It's definitely not (C) right-angled ...
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1answer
101 views

A question about a very peculiar triangle.

If we have a triangle where the Perimeter >0 and the Area >0 , and Area=Perimeter, what special condition must the angles of this triangle satisfy for this to happen? I've done a bit of research and ...
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2answers
122 views

Triangle Condition/Abbreviated Formula

So in a triangle, where the sides are length x, y ,z. The condition of (1) x+y>z, (2) x+z>y, and (3) y+z>x must be met in order for the sides of a triangle to be met. If we sum up all the conditions, ...
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1answer
75 views

Find direction, angle or co-ord of unknown vertices using only distance?

My current issue is that I have a triangle, where I know all the line distances as well as an origin coordinate. Is there any way I can then gain the coordinates of the other vertices with this ...
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1answer
323 views

Find a point on a plane

I have three points to write the equation of a plane: assume $P_1=(x_1,y_1,z_1),P_2=(x_2,y_2,z_2),P_3=(x_3,y_3,z_3)$. I can also write the equation of this plane. I want to obtain the coordinate of ...
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3answers
271 views

$\sin{\alpha}+\sin{\beta}+\sin{\gamma}>2$ Where $\alpha$, $\beta$ and $\gamma$ are angles from an acute-angled triangle.

The problem is easy to state: Prove that $$\sin{\alpha}+\sin{\beta}+\sin{\gamma}>2$$ Where $\alpha$, $\beta$ and $\gamma$ are angles from an acute-angled triangle. I only managed to turn it into: ...
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1answer
119 views

What does relative height to the hypothenuse means?

I have to solve the next problem: Given H (relative height to the hypotenuse) and R (radius of the circle inscribed in the triangle) of a rectangle triangle, can you calculate the value of its ...
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1answer
61 views

Geometry Proof Triangles

Show that if two of the corresponding angles of two triangles are equal then so is the third. Is there a formal way to prove this? I wanted to just say in one sentence that if two angles are the ...
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1answer
83 views

Determine the exact location of the centroid?

This is my last question for the day! :P Usually I am good at math but I've been sick for over a year and am now finding it hard to concentrate. :P Triangle CDE has vertices C(-2,4), D(6,2), and ...
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0answers
142 views

Finding Areas in triangles using ratios

What theorem/theorems should be used to find the shaded area? Y and M lie on the sides Ab and Bc respectively of the triangle YMB such that AY/MI= 1/4 and O/M = 1/3. Area ABC=35 PC and QA intersect ...
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2answers
73 views

Parallelograms in triangles

if posssible, could you only give me a few theorems in order to assist me in this question. Thankyou in advance! Links to simple websites would also be appreciated. In triangle $ABC$ $F$ is midpoint ...
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1answer
57 views

Proving similar triangles

In trapezium $ABCD$, $AB$ is parallel to $DC$. The diagonals $AC$ and $BD$ intersect at $X$, and $XY$ is constructed parallel to $AB$, intersecting at $X$, and $XY$ is constructed parallel to $AB$, ...
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1answer
91 views

Finding the exact area of a trapizium using similar triangles

IN the trapezium ABCD, the diagonals intercept at M. Let AM= a, BM= b, Cm = c and DM = d, and let Angle AMB be $\theta$. a=6 b=4 c=3 d=2 AB=8 DC=4 $\cos(\theta) = -\frac{1}{4}$ and $\sin ...
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1answer
362 views

Proof involving angle bisector in an arbitrary triangle

In the above figure, AD is a bisector angle A (angle BAC). How do I prove in a triangle ABC of any dimensions that, $AB > BD$ $AC > CD$ Is it also possible to prove that, $AB > AD$ ...
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1answer
123 views

Constructing an equilateral triangle from an arbitrary triangle by shifting towards an interior point

Suppose $\triangle ABC$ has no angles greater than or equal to $120^{\circ}$ and let $P$ be any arbitrary point inside $\triangle ABC$. Let $\overline{AP}, \overline{BP}, \overline{CP}$ be the line ...
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1answer
960 views

Proof of ASA , SAS , RHS , SSS congruency theorem

I have tried searching in many places for some good proofs of these theorems but couldn't find them anywhere . Even my math teacher cannot explain it to me and says that these theorems just work. I ...
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2answers
188 views

How to prove point A belongs to line t?

I'm stuck at trying to prove that any point $A$ will belong to line $t$ if and only if segments $AB=AC$, where $B$ and $C$ are symmetrical points to the line $t$ and $M$ is the midpoint of segment ...
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1answer
284 views

GRE triangle area question

I dont understand why AE is 1 if AD is 4 and the ratio between CD and AB is 9/3 or 3
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2answers
194 views

Question on triangle with heights

Prove that there exists no triangle with heights 4,7, and 10 units. I am completely puzzled.
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1answer
90 views

Right triangle with equal permeter and height - how to find side lengths?

The question: Suppose there is a right triangle with sides $a$ and $b$ and hypotenuse $c$. Its perimeter is the same as its area, and $b = 6$. What are its side lengths? I just cannot figure ...
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2answers
136 views

Finding value of an angle in a triangle.

I'm solving some practice problems to prepare for a competitive exam . Here is one which I'm trying to do for some time but still haven't found a solution to : "In the given figure , ∠ABC = 2∠ACB and ...
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2answers
144 views

Trigonometry? Get the “half” of a triangle from hypotenuse and cathetus

I've only got the following parts of a triangle: Line A to B Line B to C And optionally the Line from A to C if needed? I'm trying to get the point X Now the problem is, i've got absolutly no ...
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1answer
151 views

A problem related to area of triangles.

I'm solving some practice problems to prepare for a competitive exam . Here is one which I'm trying to do for some time but still haven't found a solution to : " In $ΔABC$ , $E$ and $F$ are such that ...
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0answers
158 views

About the area of the region where the paper is twofold when you double a piece of paper in the shape of a triangle.

Suppose that you have a piece of paper in the shape of a triangle $ABC$ whose area is $S_0$ and that the area of the region where the paper is twofold when you double the paper in two along a line is ...
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4answers
123 views

How would you measure a right triangle with sides of 1 and root 2?

This may be a silly question, but I saw this diagram on wikipedia and was intrigued: https://en.wikipedia.org/wiki/File:Square_root_of_2_triangle.svg How would such a triangle work in real life? ...
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2answers
280 views

Confusing angle-chasing question

AB = BC = CD = DE = EF = FG = GA Find angle GAB. Please, I want the correct answer. I know how to solve it, but I am getting confused by the number of triangles in it. I am getting different ...
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3answers
257 views

Proving a point inside a triangle is no further away than the longest side divided by $\sqrt{3}$

Problem: In a triangle $T$ , all the angles are less than 90 degrees, and the longest side has length $s$. Show that for every point $p$ in $T$ we can pick a corner $h$ in $T$ such that the ...