For questions about properties and applications of triangles

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0
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2answers
581 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.
12
votes
4answers
21k views

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 ...
4
votes
2answers
892 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 ...
2
votes
1answer
105 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 ...
0
votes
1answer
70 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 ...
0
votes
1answer
124 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 ...
3
votes
2answers
98 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 ...
0
votes
1answer
73 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 ...
1
vote
2answers
817 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. ...
2
votes
1answer
146 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 ...
0
votes
4answers
425 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), ...
5
votes
4answers
144 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 ...
3
votes
2answers
2k 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 ...
2
votes
1answer
118 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 ...
1
vote
2answers
189 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, ...
2
votes
1answer
89 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 ...
0
votes
1answer
344 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 ...
2
votes
3answers
374 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: ...
4
votes
1answer
132 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 ...
1
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1answer
65 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 ...
0
votes
1answer
91 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 ...
1
vote
0answers
175 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 ...
1
vote
2answers
79 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 ...
1
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1answer
59 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 $AB$, ...
0
votes
1answer
109 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 ...
2
votes
1answer
414 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, $AB > AD$ ...
1
vote
1answer
140 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 ...
0
votes
1answer
1k 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 ...
1
vote
2answers
265 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 ...
0
votes
1answer
360 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
8
votes
2answers
195 views

Question on triangle with heights

Prove that there exists no triangle with heights 4,7, and 10 units. I am completely puzzled.
1
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1answer
106 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 ...
2
votes
2answers
171 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 ...
0
votes
2answers
165 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 ...
2
votes
1answer
171 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 ...
2
votes
0answers
160 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 ...
0
votes
4answers
137 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? ...
3
votes
2answers
334 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 ...
3
votes
3answers
308 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 ...
15
votes
2answers
454 views

Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?

Let's call a point $P$ which satisfies the following condition 'a rational point'. Condition: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ ...
0
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2answers
16k views

Length of hypotenuse using one side length and angle

I bet this question has been asked a million times, but I can't find a straight answer. I need to find the length of the hypotenuse in a triangle where I have one side and all the angles. Example: ...
1
vote
3answers
2k views

A problem on a triangle's inradius and circumradius .

I'm trying to solve the following problem : In $△ABC$, $AB = AC, BC = 48$ and inradius $r = 12$. Find the circumradius $R$. Here is a figure that I drew : ( note : it was not given in the question ...
1
vote
1answer
333 views

Circumcenter coordinates for a isosceles triangle

I'm back, wow, twice a day nowadays. I need to calculate circumcenter coordinates (or at least I hope they're called that) at point C for an isosceles triangle (the circle must be such, that created ...
2
votes
1answer
486 views

Calculate angle of triangle

I need to calculate the angle between two sides, I have the length of A & B sides, but don't know how to find the angle... Both sides are the same length. I can get the start and end vectors of ...
1
vote
2answers
63 views

relation between inscribed and circumsdribed circle

Let T be the triangle with side lengths $b_1,b_2,b_3$, and $r_{insc}$ and $r_{cir}$ be the radii of the inscribed and circumscribed circles,respectively, I need to show that $$ \frac ...
0
votes
0answers
89 views

Fastest way to check whether the triangle inequality is satisfied

If we are given the lengths of the three sides of a triangle, and we simply add the 2 smallest sides and check to see if the sum is larger than the third side, will this always yield the correct ...
5
votes
3answers
446 views

Prime Number in triangle

I had a question here, the measures of the sides of a right triangle (a single unit) can be prime numbers? If they can not, why?! But, if you can, could you help me find an example?
3
votes
1answer
183 views

the ratio of the following two areas

Suppose you have the following triangle $ABC$: with the following properties: $|AB|=4\cdot |AA'|$, $|AC|=4\cdot |CC'|$, $|BC|=4\cdot |BB'|$. I have to find the ratio of the total area of the triangle ...
6
votes
2answers
142 views

Find The range of $r/R$.

Given a triangle $ABC$ with angle $A=90^{\circ}$. Let $M$ be the midpoint of $BC$. If the inradii of the triangles $ABM$ and $ACM$ are $r$ and$\ R$ respectively, then find the range of $\dfrac rR$ .
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vote
4answers
130 views

How to mathematically define “on the outer side of the triangle”?

Given the coordinates of a triangle's vertexes, I'm trying to find its Fermat point programmatically. In one step of the algorithm that I'm trying to implement, I have to draw equilateral triangles on ...