For questions about properties and applications of triangles

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1answer
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How would I find the area of a triangle given three sides and using either the sine/cosine laws?

Triangle ABC has sides $8.5m$ (a), $7.1$ (b), and $9$ (c). I have been asked to find the area of the triangle using trigonometry.
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2answers
166 views

Solving for the length of a side of a triangle

I have a problem in which I'm supposed to solve for the length of the two sides of the triangle below. I assumed that it would simply boil down to $x+5=\sqrt{4x+52}$, and converted to standard form, ...
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2answers
714 views

Combinatorics. Inscribed Triangle in a decagon. No shared sides.

How many different triangles can be inscribed inside a regular decagon such that the triangle shares its vertices with the vertices of the decagon, but the triangle shares none of its sides? Here is ...
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1answer
46 views

10.5“ and 32” hypotenuse, a=8.5, b=42.5, what angles are the 10.5“ and 32” Hypotenuse?

I have a ramp that has a concave "kink" in the angle. The first length of the hypotenuse is 10.5", the next is 32". The triangle is 8.5" tall (a) The triangle is 42.5" long (b) How do I figure ...
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1answer
204 views

points inside square that form a triangle

the following question beat me. How from given any 9 points inside a square of side 1 we can always find 3 which form a triangle with area less than $1/8$ .
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4answers
41k views

Given the base and angles of an isosceles triangle, how to find length of the two sides?

I can't seem to find a textbook solution to this. It is always assumed that the length of the sides is know. Isolceles triangle So the base $a$ is known. The bottom angles where $\alpha$ and the ...
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2answers
153 views

As shown in the figure: Prove that $X=30.$ [closed]

Any idea about this problem: As shown in the figure: Prove that $X=30.$
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2answers
1k views

Can every triangle be divided into five isosceles triangles?

That's my problem: Can every triangle be divided into five isosceles triangles? I've got to give evidence why this is true or not true... (sorry for possible language mistakes - I'm from Germany) ...
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10answers
11k views

What's a proof that the angles of a triangle add up to 180°?

Back in grade school, I had a solution involving "folding the triangle" into a rectangle half the area, and seeing that all the angles met at a point. However, now that I'm in university, I'm not ...
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2answers
614 views

3d geometry: triangle 2 points known, find 3rd point

I have a 3d triangle ABC. Lengths AB, BC, and AC are known. Coordinates of points A and B are known. Point C only the y value of the coordinate is known. I believe there are 2 points that can satisfy ...
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1answer
148 views

High School Geometry - If $BC$ is the greatest side of $\triangle ABC$, $D$ & $E$ are points on $BC, CA$…

If $BC$ is the greatest side of $\triangle ABC$, and $D$ & $E$ are points on $BC$ & $CA$, respectively, prove that $BC \ge DE$. Clearly, equality holds iff $D$ is on $B$ and $E$ is on ...
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How do I show that the pythagoras theorem holds for the specific case of an “isosceles right triangle”?

Figure shows a rectangle $ABCD$ and an isosceles triangle $\triangle DEC$. $AD=BC=z$;$AB=DC=y$;$DE=CE=x$ One solution is as follows. We know that the pythagoras theorem holds for a right triangle ...
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1answer
2k views

Finding the line integral around a triangle

How can I determine $\int xy \;ds$ of a triangle with points $(0,0)$, $(1,0)$ and $(1,1)$ *The integral has the letter $C$, which I am not sure how to input here. I know it may seem easy, but I am ...
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4answers
733 views

How can every triangle have a circumcircle

Let's take for example $\triangle ABC$ with $\angle A = \angle B = 1^o$. How can a triangle like this have a circumcircle? My confusion is with triangles like this in general, with very long sides.
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3answers
238 views

Does every set of any three vertices of a cube determine a right triangle?

I recently came across this in my textbook: Any three vertices of a cube determine a right triangle. Is this a true statment? My initial thought was that is was, but the answers say otherwise. ...
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2answers
111 views

Calculating meeting point where line intersects arch

How do I find the point $p$ where the arch meets the red line if the angle of the blue are is known and the height of the yellow?
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4answers
833 views

Calculating an angle adjacent to hypotenuse given two points

I'm working on a chapter in my book dealing with touch input, and my memory of high school trig (from circa 1988) is failing me. My search here has not yielded anything that I'm capable of applying to ...
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3answers
76 views

Is this triangle possible to draw?

If the triangle must have the following sides: DE = 3cm DF = 8cm EF = 4cm And not taking into consideration anything about the angles. Is it possible to draw such a triangle?
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0answers
151 views

Determining a point in 3D space

So given a point, a rotation around the y-axis, a rotation around the x-axis, and a distance, how can one calculate the relative point in space? For example, the beginning coordinates are (0,0,0). ...
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1answer
166 views

Triangle inequalities, with angle bisector

I came across this question while I was taking one of the pratice Mu Alpha Theta tests for my school and I wasn't sure how to solve it. It reads: In $\Delta USA $, $\angle S$ is bisected by ...
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1answer
251 views

Squares in a triangle?

I've got some trouble... IJKL is a square and B, I, J, C are aligned (alternatively, |IJ| is confounded with |BC|. h is the height of acute $\triangle$ ABC from A to side BC. C1 is the red ...
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1answer
93 views

a triangle problem of angles

suppose in triangle ABC , angle of BAC is 60 degree. if K is intersection point of [CM] median(for segment[AB] )and [BN] altitude. also suppose |KM|=1 cm and |CK|=6 cm calculate angels of triangle ...
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2answers
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Minimizing (and maximizing) the area of triangles

How would one solve questions like this one here in general? I have gotten an answer for that question, but I don't understand what's the intuition behind it. Can smoeone clearly explain how to ...
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4answers
2k views

Find an angle in a given triangle

$\triangle ABC$ has sides $AC = BC$ and $\angle ACB = 96^\circ$. $D$ is a point in $\triangle ABC$ such that $\angle DAB = 18^\circ$ and $\angle DBA = 30^\circ$. What is the measure (in degrees) of ...
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1answer
4k views

Compute the length of an equilateral triangle's side given the area?

Given the area of an equilateral triangle, what is an algorithm to determine the length of a side?
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3answers
369 views

On Ceva's Theorem?

The famous Ceva's Theorem on a triangle $\Delta \text{ABC}$ $$\frac{AJ}{JB} \cdot \frac{BI}{IC} \cdot \frac{CK}{EK} = 1$$ is usually proven using the property that the area of a triangle of ...
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1answer
228 views

Altitude of tetrahedron?

I'm really curious to know any relationships between the altitude of a tetrahedron and how the foot of this altitude splits the base triangle. For example if you have a tetrahedron PABC with apex P, ...
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2answers
326 views

Triangle in hexagon

In a regular hexagon ABCDEF is the midpoint (G)of the sides FE and S intersection of lines AC and GB. (a) What is the relationship shared point of straight ...
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1answer
76 views

Solution for the value of angle $A$ of a triangle

In triangle $\triangle \; ABC$ , if $$2\frac{\cos A}{a} + \frac{\cos B}{b} + 2\frac{\cos C}{c} = \frac{a}{bc} + \frac{b}{ca}$$ find angle $A$. This is a quiz bee problem sent to me by my friend in ...
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2answers
548 views

Construct a triangle given one side, its height and inradius

I've been scratching my head with this problem: "Draw a triangle given one of its sides, the height of that side and the inradius." Now, I can calculate the area and obtain the semiperimeter. From ...
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4answers
2k views

what's the name of the theorem:median of right-triangle hypotenuse is always half of it

This question is related to one of my previous questions. The answer to that question included a theorem: "The median on the hypotenuse of a right triangle equals one-half the hypotenuse". When I ...
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3answers
194 views

Prove that point M is on circle c

It's hard to create question names that make sense. Anyhow, the following is another question from my math assignment. Line-segment AB has a fixed length of 10 units. point A moves on the positive ...
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1answer
6k views

How to calculate radius when I know the tangent line length?

For my math homework, I was asked this question: The tangent lines from O hit a circle with center M and radius r in R and S. Calculate r. -The length of OR and OS is 4 How do I calculate the ...
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2answers
2k views

How do you find the equation for the angle bisecting line given three coordinates that make up an angle?

I have three points,$$A =[A_x,A_y]\,,\, B =[B_x,B_y]\,,\,C =[C_x,C_y]$$ How could one calculate the equation for the line that bisects the angle $\,\angle ABC\,$ (eg., passing through $B$)?
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Does the angle bisector always pass through the midpoint of any line segment between the two sides of the angle?

Consider this image: will the angle bisector of angle AOB always pass through the midpoint of AB, regardless of the lengths of AO and BO?
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3answers
81 views

Prove that altitude² = pq?

The following is a question for my math class. I just cannot figure it out. Given is that: h is the altitude that divides the longest side of this right triangle into p and q. Question: Prove that ...
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1answer
544 views

Integer solutions to linear equation – Triangle with set perimeter

We have a triangle with the sides a, b and c where: ...
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1answer
447 views

Area of triangle ABC inside circle

Consider the following diagram: $AB+AD=DE$, $\angle BAD= 60$, and $AE$ is $6$. How do we find the area of the triangle $ABC$?
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1answer
171 views

find distance from point in circle to perimiter

If I have the following circle, with centre in red and a random point in the circle in blue. I know the radius ,r, length of d, and the angle p: I then create a a new green point and I know the ...
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2answers
112 views

Triangle angles

How would I prove that, in any triangle, any of the exterior angles is bigger than any of the remote interior angles? Help would be much appreciated!
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0answers
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How to prove this relationship between sides length in the specific triangle? [duplicate]

Possible Duplicate: Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle When I was exploring a web collection with geometrical problems I found this one: How can I prove that in ...
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2answers
123 views

Angular radius of a sphere

Given a sphere with radius $r$ about a point $c$, what's the apparent angular radius $\alpha$ of that sphere from point $P$? In other words, if $\vec{o} = c - P$, what's the maximum angle another ...
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2answers
208 views

Geometry - Area of Siamese Triangles

How can I find the Area of this figure? It is quite curious because it is a particular case of this sequence: Anyone know how to find the area of this sequence as a function of the number of ...
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0answers
88 views

Number of distinct centroids of triangles formed by 40 equally spaced points on a the perimeter of a square

Suppose that we are given 40 points equally spaced around the perimeter of a square, so that four of them are located at the vertices and the remaining points divide each side into ten congruent ...
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2answers
169 views

Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle

How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
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2answers
98 views

Difference between $\angle ABC = 90^o$ and $\angle B = 90^O$

When you have a random triangle $\triangle ABC$, what exactly is the difference between $\angle ABC = 90^o$ and $\angle B = 90^o$? In which cases is it the same, in which cases is it different? What ...
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3answers
96 views

Calculate incircle radius.

A circle is inscribed in a right angled triangle ABC where AC is the hypotenuse. The circle touches AC at point P. Length of AP = 2unit and CP = 4 units. What is the radius of the circle?
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4answers
880 views

Constructing a triangle given three concurrent cevians?

Well, I've been taught how to construct triangles given the $3$ sides, the $3$ angles and etc. This question came up and the first thing I wondered was if the three altitudes (medians, ...
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1answer
374 views

Prove that the Simson line of $P$ bisects the segment $HP$ from the orthocentre $H$ to $P$

Let $ABC$ be a triangle with orthocentre $H$ and circumcircle $\odot(ABC)$. Suppose $P\in\odot(ABC)$. Let $\gamma$ be Simson's line of $P$ wrt $ABC$. Prove that $\gamma$ bisects $PH$.
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1answer
151 views

Inequality in Triangles

Given is a triangle on points x,y,z in the plane. This triangle has two points a and b on different sides. I would like to show that the following inequality has to hold: $\max \{d(b,x), d(b,y), ...