For questions about properties and applications of triangles

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2answers
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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
40 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
59 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
44 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
58 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
40 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 ...
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1answer
59 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
80 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
87 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
43 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
461 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
37 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 ...
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1answer
91 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 ...
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1answer
24 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
100 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 ...
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1answer
79 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
35 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
45 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 ...
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0answers
17 views

Rationalizing triangle relationship for a bar inside a hemisphere

Find ratio of length AE to diameter of the hemisphere Given that ABD = 90 degree, AO = BO, O is the center of the sphere. This is actually a physics problem, but I bump into geometric problem ...
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0answers
24 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 $AX/XC=CY/YB=XZ/ZY$, prove that the area of triangle $ABC$ is given by: $$ ...
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0answers
22 views

Rotating a triangle in different coordinate systems.

My android application uses openGL. OpenGL coordinate system has the origin in the middle and goes from -1 to 1. When I am rotating an equilateral triangle in the openGL coordinates, the triangle ...
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1answer
28 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
46 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
39 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
67 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 ...
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1answer
32 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
77 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
37 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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0answers
23 views

Solving ray/triangle intersection - comparison on methods

In 3D I have a ray $q + t\vec{d}$, and a triangle with vertices $a$,$b$,$c$. I want to find the parameter $t$. I have solved this by noting that a vector in the triangle plane is perpendicular to the ...
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0answers
129 views

Finding general Cartesian coordinates of the third vertex of a triangle lies between two circles

I'd like to find the Cartesian coordinates of the vertices $(a, b, c)$ of the triangle $T$ inscribed in the circle $S^1$ and circumscribed about the circle $D$ ? I start my calculations as follows: ...
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2answers
39 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
43 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
26 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
25 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 ...
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2answers
33 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 ...
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3answers
253 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 ...
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2answers
84 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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2answers
83 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
105 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 ...
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4answers
78 views

Calculate the angles of a isosceles triangle

In the triangle below, is there a way to calculate the $x$ and $y$? To be more specific, $b = 12.8\rm\,cm\ $ and $h = 10\rm\,cm$, hence $a = 11.87\rm\,cm$. I don't know what to do from here.
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1answer
42 views

A geometric inequality

Let $M$ be a point inside the triangle $ABC$. $AM$ intersects the circumcircle of $MBC$ for the second time at $D$. Analogously define $E,F$. Prove the following : $$ ...
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1answer
27 views

$PC+PD$ is least when the angles $CPA$ and $DPB$ are equal

$C$ and $D$ are two points in the $same$ side if a straight line $AB$ and $P$ is any point in $AB$. Show that $PC+PD$ is least when the angles $CPA$ and $DPB$ are equal No idea how to solve this ...
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1answer
60 views

Length of a segment on right triangles that share same hypotenuse

I have two right triangles that share the same hypotenuse. Can the length of Xb be found using just the other lengths shown (X, L, Y)? I have only been able to find it by using a combination of the ...
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0answers
46 views

Rationality in Triangle

How can I justify this answer? I think the answer is infinite, but cannot justify it///
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1answer
48 views

Circle theorem/triange angle question

I am doing practise papers and there is one question I cannot understand even with the mark scheme. I have added the pictures below: Question (with added annotations): Mark scheme: The question ...
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1answer
41 views

Is the given triangle unique?

I was reading Polya's How to Solve It when I came across the following problem. Construct a triangle with an angle, the length of altitude through that angle and the perimeter of the triangle given. I ...
2
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1answer
52 views

Hyperbolic Triangles and Uniform thinness

My textbook states that all triangles in hyperbolic space are uniformly thin in the following way: If $ABC$ is a triangle and $x$ is a point on one side, then there exists a point $y$ on one of the ...
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2answers
45 views

Length of a line in an isosceles triangle. (mind boggling )

In an isosceles triangle ABC, side AB and AC are equal in length. There exists a point D on the side AB. The angle BAC is theeta . The side AD is two units smaller than AC .What is the generalized ...
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2answers
258 views

Maximal area covered by two triangles in unit circle

What is the maximal area covered by two triangles in a unit circle? There are no restrictions other than that. They can overlap, touch the circle, not touch the circle etc. So far I have shown In ...
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3answers
107 views

find angle sine knowing all sides

I know all the sides of an arbitrary triangle but not the angles, and I want to find the sine of any angle. ...