For questions about properties and applications of triangles

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1answer
93 views

Find points of triangle, one point, all sides and all angles known

Imagine the setup above; how can I calculate the points P1 and P2 if all angles, all sides A,B,C and point P3 are known?
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1answer
97 views

Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
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3answers
190 views

Area of a triangle adjacent to two similar triangles

This GMAT problem states: In the figure above $AD = 4$, $AB = 3$ and $CD = 9$. What is the area of triangle $AEC$ The solution states: to find the base we need to see that triangles $AEB$ and ...
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1answer
103 views

Minimum Value of $x_1+x_2+x_3$

For an Acute Triangle $\Delta ABC$ $$\begin{align}x_n=2^{n-3}\left(\cos^nA+\cos^nB+\cos^nC\right)+\cos A\,\cos B\,\cos C\end{align}$$ Then find the least value of $$x_1+x_2+x_3$$ My Approach: I have ...
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1answer
367 views

Rotation matrix of triangle in 3D

How can I find out the rotation matrix of a right angle triangle defined by 3 points in 3D space (assuming the un-rotated triangle faces the x axis)
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0answers
52 views

Find angle of an arc in the circle using 3 coordinates

I want to find angle of semicircle. I have 3 coordinates (center_a,center_b) , (pivot_a,pivot_b) and (last_point_a, last_point_b). From triangle , i can find angle using equation using cosine ...
2
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1answer
56 views

Area of a triangle whose each side is less than 2 and greater than1.

What is the area of a triangle if each of its sides is greater than 1 and less than 2? My Try:Let a,b,c be the sides of triangle,then ...
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3answers
38 views

Locus Similar Triangle

ABC is a triangle and XY is variable straight line parallel to AC meeting BC and BA in X, Y respectively. If AX and CY meet at P find the locus of P.
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1answer
58 views

If you know 2 sides of the triangle, wha is the third side?

I understand why A & C are correct but I don't get how E is a possible length since whatever number I plug in for x I get a number greater than 5x+5...
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4answers
224 views

How many pieces of information are needed to determine a triangle?

Typically 2 sides and 1 angle need to be given in order to determine a unique triangle. Alternatively 1 side and 2 angles, or the Cartesian coordinates of three vertices, or the area, base, and ...
1
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1answer
111 views

Probability distribution of the third side in triangle

Given the two distributions of two sides of a triangle (for example, Uniform and Rayleigh) and the distribution of an angle between them (Uniform[0,Pi]), find the length of the third side. What i ...
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2answers
294 views

Triangle problem - finding the angle

Please look at the following figure: All the angles are in degrees. I have to find $x$. I am really no good at solving geometry problems. I tried to search the internet for similar problems and ...
2
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1answer
59 views

Ratio of sides of triangle $ABC$

If in a triangle $\Delta ABC$ with $a$, $b$ and $c$ as sides $$\begin{align}\left(Cot\frac{A}{2}\right)^2 ...
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2answers
365 views

A triangle with integer co ordinates and integer sides

Is there a triangle with integer sides as well as integer co ordinates when none of the angles is $90$? I tried to solve the general case but I am stuck with it. Update: Let the Triangle be $T$ ...
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2answers
46 views

Determining the coordinate of C to minimize the area of a triangle ABC

Given $A=(0,-10)$ and $B=(2,0)$. Determine the coordinate of $C$ in the curve $y=x^2$ which minimalize the area of triangle $ABC$.
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1answer
102 views

How to prove the property of the Lemoine point of a triangle?

From Wolfram MathWorld, I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum. How to ...
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1answer
533 views

How can I transform a 3D triangle to xy plane

Suppose I am given a triangle ABC and its corresponding vertex coordinates in 3D. I want to transform ABC in such a way so that vertex A lies on global (0,0,0) coordinate, B lies on (dist, 0, 0) ...
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2answers
114 views

Locus of the Orthocenter of the Traingle

Coordinates of $\Delta ABC$ are $A(3,4)$, $B(5 \cos\theta, 5 \sin\theta)$ and $C(5 \sin\theta,-5 \cos\theta)$. Find the locus of its orthocenter. My idea: It is clear that $(0,0)$ is equidistant ...
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2answers
64 views

A triangle problem

In a triangle, the sum of two sides is $x$ and the product of the same two sides is $y$. If $x^2 - c^2=y$ where c is the third side, then what is the ratio of the inradius to the circumradius of the ...
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1answer
62 views

Geometric proof with a isosceles triangle

Given is $\triangle ABC$ with the medians $AD$, $BE$ with $|AD|=|BE|$. The medians intersect in $S$. a. Use similar triangles to show that $|AS|:|SD|=|BS|:|SE|=2:1$. b. Prove that $\triangle ABC$ is ...
3
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1answer
79 views

Paths followed by Morley triangle vertices as apex moves parallel to base

Let the vertices of a triangle $T$ be $(A,B,C)$, and $(a,b,c)$ the vertices of its Morley triangle $M$. Designate vertex $C$ as the apex of $T$. Now move apex $C$ parallel to $AB$, all the while ...
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1answer
299 views

Triangle in 3D space point X and Y coordinate know find Z

I have a triangle in a 3D space. I know the points X an Y coordinate but I dont know the Z. How can the Z be calculated by knowing the points of the triangle and the X an Y coordinate of the point ...
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3answers
252 views

Finding an area of a triangle inside of a triangle, given certain areas of other triangles, and area ratios.

I'm studying for the Waterloo Math Contest (Galois, Gr. 10) taking place in April of 2015 and I am preparing by looking at previous problems and solving them. This is question 4(c) on the 2010 Galois ...
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2answers
50 views

How to find length of the sides of a triangle given the ratio of the sines of the sides?

Consider $\triangle ABC$. Let $\dfrac{\sin A}{\sin B} = \dfrac56$ and $\dfrac{\sin B}{\sin C} = \dfrac45$. Find $\dfrac{\vert AC\vert\cdot \vert AB\vert}{\vert BC\vert}$. If there is no definite ...
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2answers
471 views

Solving for Cos Exactly

How to solve $\cos(\dfrac{5\pi}{6})$ and $\cos(\dfrac{7\pi}{6})$ exactly? I couldn't use special triangles to solve this either.
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2answers
62 views

Get the angle in a circle using center, radius and one point in a circle.

There is a circle and i know Point1 this is fixed and i know another point Point2 which can be anywhere in the circle. and i want to know the angle which is made at center. Thanks Your help will be ...
1
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1answer
202 views

How to calculate a variable vertex's coordinates on a scalene triangle given an original triangle

The vertex I'm looking for lies on one of the altitudes of the red triangle which we know everything about via calculation. Given the desired, final angle (135 degrees, but theoretically, any ...
0
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1answer
52 views

Area of triangle on a sphere (not spherical triangle)

How do I find the area of a triangle on a sphere, and the triangle is not a spherical triangle, for example, the triangle is formed with two geodesics and a line of latitude. Is there a specific ...
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4answers
115 views

Finding $\sin^{-1}(x)$ without using a calculator

I don't understand how to compute $\sin^{-1} (0.6293)$, to figure out the angle without using a calculator. I understand how to find the answer if I use a calculator but I don't understand the ...
0
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1answer
503 views

How many triangles are see in complete K5 graph

How many triangles are on picture below? On yahoo answers I have found that numbers of triangles in complete graph with n nodes is: $\frac{n(n-1)(n-2)}{6}$ But how this formula has been estimated? ...
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2answers
76 views

Find length of $CD$ where $\angle BCA=120^\circ$ and $CD$ is the bisector of $\angle BCA$ meeting $AB$ at $D$

$ABC$ is a triangle with $BC=a,CA=b$ and $\angle BCA=120^\circ$. $CD$ is the bisector of $\angle BCA$ meeting $AB$ at $D$. Then the length of $CD$ is ____ ? A)$\frac{a+b}{4}$ B)$\frac{ab}{a+b}$ ...
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1answer
209 views

Finding the area of a triangle in terms of the radius of the excircle

Prove that the area of a triangle $ABC$ is $$\frac12 (b + c - a)r$$ where $r$ radius of the excircle opposite to $A$ and the rest of the symbols have their usual meaning. I started off with the ...
2
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2answers
172 views

A question about 4 concyclic points

In a triangle $ABC$, let $I$ denote its incenter. Points $D, E, F$ are chosen on the segments $BC, CA, AB$, respectively, such that $BD + BF = AC$ and $CD + CE = AB$. The circumcircles of triangles ...
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0answers
30 views

area of triangle in terms of sides ratio [duplicate]

In $\triangle ABC$, $X$ and $Y$ are points on the sides $AC$ and $BC$ respectively. If $Z$ is on the segment $XY$ such that $\frac{AX}{XC}=\frac{CY}{YB}=\frac{XZ}{ZY}$, prove that the area of ...
1
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1answer
59 views

Area of triangle in a different coordinate system.

This is for an android application but I think it is too mathematical to put on normal SO. I have a coordinate system where the origin is (0, 0), and the x and y axis go from -1 to 1. This coordinate ...
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2answers
57 views

Proving in a triangle

$AB$ and $CD$ are two straight lines intersecting in $O$. $XY$ is another straight line. Show that in general two points can be found on $XY$ which are equidistant from $AB$ and $CD$. But isn't ...
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1answer
48 views

Prove that $\sin A - \sin B + \sin C = 4\sin A/2 \cos B/2 \sin C/2$

Prove that $\sin A - \sin B + \sin C = 4\sin A/2 \cos B/2 \sin C/2$ occurs in an $ABC$ triangle. I don't know how to solve the RHS... Can anyone help me please?
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0answers
132 views

Calculate height from two right angled triangles sharing an edge

I am trying to calculate the perpendicular distance of a unicycle-like robot from a wall using two successive measurements from an ultrasonic sensor. The problem is modelled as shown: (EDIT). The ...
0
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1answer
45 views

Prove inequality in a triangle

Prove that In a triangle having sides $a, b, c$ $$a^2(b+c-a)+b^2(a+c-b)+c^2(a+b-c)\le 3abc.$$ I tried using the basic two side sum greater than third property but got nothing hope you guys help ...
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2answers
177 views

Integer Triangles with Perimeter $n$

How many triangles are possible with positive integer side lengths for perimeter $n$? My attempt so far has been bashing for $n=1,\; 2, \cdots , 13$ and calculating how many triangles are ...
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1answer
111 views

Juxtapose two triangles with a common edge

I'm not experto in geometry but I'm trying to do a software that handle triangles in various way. And I'm trying to learn geometry, of course : ) I have one fixed triangles $T1 = \hat{ABC}$ and a ...
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2answers
45 views

Geometry, two perpendicular lines

"Let $\hat{ABC}$ be an isosceles triangle with $AB=AC$. $D$ is a point on $BC$ such that $DC=DB$ (middle of $BC$). $E$ is the projection of $D$ on $AC$ and $F$ the middle of $DE$. Prove, using vectors ...
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0answers
100 views

minimum sum of distances from vertices

Find a point on the plane of a triangle such that the sum of its distances from three vertices is minimum. I am guessing that it is the centroid but I can't prove that.
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2answers
43 views

Can the vertex angle of an isosceles triangle be found without the law of cosines (no calculator)?

If we know three sides of an isosceles triangle, can we find the measure of the angles without using a calculator (that means no law of Cosines/Sines).
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1answer
45 views

Finding a side of a triangle with one side, angle and a quotient of two other sides.

Solving a firing-with-prediction puzzle in the game I am developing I found myself looking on the internet of solutions about triangle and its side. It turns out there is very few information about ...
4
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2answers
49 views

formula for number triangles

Hi, I have a triangle starting from $0$ and going up by one on the bottom row until there are $r$ items on the bottom row and there are $r$ rows a number is formed by adding the two numbers towards ...
5
votes
3answers
501 views

Need algebra tip about $a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$ for sides of a triangle

I just got a long expression: $$a^4 + b^4 + c^4 - 2b^2c^2 - 2a^2b^2 - 2a^2c^2$$ and I need to prove its less than zero for every $a$, $b$, and $c$ which are triangle sides I really need tips how to ...
2
votes
2answers
264 views

Equal perimeters of squares and right angled isosceles triangles

Consider a square ABCD having length l and breadth. Now start folding the sides AB and AC so that the figure becomes something like this $$$$ All the vertical and horizontal folds/stairs are equal in ...
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1answer
910 views

Cut A Shape Into Two Triangles

I have this shape: , and I want to put a straight line somewhere through the shape to cut it into two triangles. I know that this is possible, but I don't know how. Any help is appreciated!
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2answers
840 views

Number of triangles in a graph based on number of edges

Given a graph $G(V,E)$, what is the maximum number of triangles that this graph can have in terms of $|E|$? I know that there is a triangle listing algorithm that lists all the triangles in ...