For questions about properties and applications of triangles

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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
134 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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2answers
144 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 ...
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1answer
52 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
54 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
42 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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109 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
43 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
123 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
92 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
43 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
69 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
37 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
34 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
46 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
368 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
223 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
753 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
489 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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5answers
186 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
50 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
28 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
77 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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1answer
60 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
129 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
44 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 ...
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1answer
68 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
59 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 $AB$. $\angle BAC$ is $\theta$. The side $AD$ is $2$ units smaller than $AC$. What is ...
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2answers
416 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
120 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. ...
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3answers
41 views

Is this triangle question missing information?

In the $\Delta KLP$, find $a+b$: My question is that: isn't some information missing from the question? Because all I can see is is that $ \usepackage{ gensymb } \angle SKP = \angle LTS = ...
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1answer
62 views

To prove inequality for two similar triangles $ABC$ and $A_1B_1C_1$ given that $A_1B_1C_1$ is inscribed in $ABC$

Consider a triangle $ABC$. A directly similar triangle $A_1B_1C_1$ is inscribed in the triangle $ABC$ such that $A_1,\;B_1\;,C_1$ are the interior points of the sides $AC,\;AB\;and\;BC$ respectively. ...
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1answer
172 views

Find Coordinates on a track

Charlie and Alexandra are running around a circular track with radius 60 meters. Charlie started at the westernmost point of the track, and, at the same time, Alexandra started at the northernmost ...
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2answers
46 views

Finding coordinates on a circle

So this problem I am have difficulty with. I think where I am going wrong is how to calculate the initial theta. Do I just use pi/2 because in the pictures it show to angle theta off the 90 degree ...
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1answer
122 views

Find the height of the dam given angles of a triangle

The top of a dam has an angle of elevation of 1.3 radians from a point on a river. Measuring the angle of elevation to the top of the dam from a point 155 feet farther downriver is 0.8 radians; assume ...
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1answer
29 views

Finding the ratio of a dissected isosceles?

I have trouble trying to find relationship between sub-triangles.
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3answers
41 views

Find the number of positive integers $b$

Let $a, b$ be positive real numbers such that $10 < a < b$. Then find the number of positive integers $b$ such that (i) $10, a, b$ are in geometric progression, and (ii) $10, a, b$ form the ...
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1answer
49 views

Proving by using inequality of triangle

suppose that points a and b are from different sides of a line m. Find a point y on line m such that the absolute difference of the YA and YB is maximal. Show proof.
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2answers
49 views

Sum of areas are equal

Given an equilateral triangle $(ABC)$ and let $P$ be an arbitrary point inside this triangle. Moreover let $V,W,T$ be the orthogonal projections of the point $P$ on to the sides $(AB), (BC), (CA)$ ...
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1answer
50 views

Length of sides and type of triangle [closed]

If I have the length of three sides, how do I figure out if it's a right triangle? So what is the formula that will help me find this out?
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2answers
57 views

Drawing a triangle with 2 known corners and all side lengths

Assume that there are three points $A$, $B$ and $C$. All the pairwise distances are known $(|AB|, |AC|, |BC|)$. But none of the coordinates are known. I want to draw a triangle using those points. ...
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2answers
49 views

Geometry basic problem

If I have a triangle with given: $b-c=3 \space\text{cm}$, $a=6\space \text{cm}$ and $\alpha$ is $30^\circ$, how do I draw this? Please help me by telling me where I can find this type of exercises ...
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3answers
265 views

Smallest square containing a given triangle

Given a triangle $T$, how can I calculate the smallest square that contains $T$? Using GeoGebra, I implemented a heuristic that seems to work well in practice. The problem is, I have no proof that it ...
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1answer
52 views

How to prove these triangle relations?

$O$ is the circumcenter of triangle $ABC$, whereas $G$ is the centroid and $H$ is the orthocenter. $R$ denotes the circumradius. How can I prove the following relations: $OH^2=9R^2-(a^2+b^2+c^2)$. ...
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0answers
77 views

Moving up the Y axis the length of the hypotenuse of a right triangle

If I have a right $\triangle ABC$ with $B$ being the right angle and length $AB = 50$ and length $BC = 50$. Based on the Cartesian coordinate system if I wanted to move up the Y axis the length of the ...
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1answer
93 views

Question about Pasch's Postulate, line going through all three sides of a triangle

I've been reading the textbook Elementary Geometry from an Advanced Standpoint by Edwin E. Moise (3rd ed.). My problem with his wording of Pasch's Postulate, and then a subsequent problem which ...
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1answer
46 views

Number Triangle pattern

I have a number triangle as follows: $$\begin{array}{|c|c|c|} \hline 0 & 0 & 0 & 1 & 0 & 0 & 0 \\ \hline 0 & 0 & 1 & 1 & 1 & 0 & 0 \\ \hline 0 ...
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1answer
80 views

How to solve this geometry question?

Let $\triangle ABC$ be an acute-angled triangle; $L$, $M$, $N$ be the feet of perpendiculars respectively from $A$, $B$, $C$ to the opposite sides; $D$, $E$, $F$ be the midpoints of the sides $BC$, ...
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0answers
57 views

Closest Points on Two Triangles in 3D Space

I have two triangles in 3D space, defined by 3 (x, y, z) points each. I'm looking to find the closest points between the two triangles, whether that be on surface, edge, or point. I'm unsure how to ...