# Tagged Questions

For questions about properties and applications of triangles

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### Prove that an Equilateral cannot have natural number points

Let $OAB$ be an equilateral triangle with $O(0, 0),\ A(m, n),\ B(x, y)$, where $m, n \in \mathbb{N}^{\ast}$ and $x, y \in \mathbb{R}_{+}$. Prove that $B$'s coordinates can't be both natural numbers....
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### General Triangles: Area, lengths and angles calculations

I have a question on General Triangles (as in non right angle). I’m trying to create a program that calculates angles and sides based on the user entering Area and some sides length or angle ...
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### Given latitude and longitude, how to find central angle and arc length of spherical triangle?

Lewis and Clark followed several rivers in their trek from what is now Great Falls, Montana, to the Pacific coast. First, they went down the Missouri and Jefferson rivers from Great Falls to Lemhi, ...
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### geometry - prove that you can make new triangle with..

I have a triangle, the length of heights are $i,h,g$. Prove that we can build a new triangle so that the lengths of the sides are: $i^{-1}, g^{-1}, h^{-1}$ (see picture)
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### Joint PDF of two random variables in a triangle

Let the random variables $X$ and $Y$ have a joint PDF which is uniform over the triangle with vertices at $(0, 0), (0, 1 )$ and $(1, 0)$. Find the joint PDF of $X$ and $Y$. So ...
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### Determine the third point in right triangle only knowing the coordinates of the other two points

I have a right triangle $ABC$. I am given the coordinates of the two points $A(x_1, y_1)$ and $C(x_2, y_2)$. Given points $A$ and $C$, I want to determine the coordinates of $B$. I know there are two ...
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### Area of non-spherical triangle on a sphere

This is a followup to the question Area of triangle on a sphere (not spherical triangle) Since it's now almost two years later, I'm making it a new question. The problem is to find the area of ...
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### Rotate right triangle with perimeter 1 about the hypotenuse [closed]

We rotate every right triangle with perimeter 1 about its hypotenuses. Is it true that we can choose a solid from so obtained solids that has maximum volume? If yes, what's the volume? I guess I ...
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### The ratio of areas of two triangles with the same altitude is equal to the ratio of their bases

How can we prove that the ratio of areas of two triangles of equal altitudes is equal to the ratio of their bases?
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### Inequality involving circumradii

Let $ABC$ be a triangle and $M$ a point on the side $BC$. Let $R_1$,$R_2$, and $R$ be the circumradii of the triangles $ABM, ACM$, and $ABC$. Show that $\max\{R_1,R_2\} \geq R\cos\frac A 2$.
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### Three Altitudes of a triangle are concurrent

I have been told that this well known fact can be shown using only Euclid's propositions from books one to three, and cyclic quadrilaterals. I can't figure out how to start, which quadrilateral ...
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### Law of Cosine formula that I can't seem to rearrange.

I was in midst of solving a trig problem, and it required using the formula of Law of Cosine. For my case, I had to solve for a specific variable which was $\cos (A)$. Would you show me step-by-step ...
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### Equilateral triangle with vertices whose coordinates on the Cartesian plane are integers. Does such a triangle exist? [duplicate]

Can you build an equilateral triangle on a Cartesian plane whose vertices only have integer values as their coordinates? Looking at the simplest example, i.e. a triangle with vertices (0,0), (1,0) ...
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### Geometry: Determining the length of a side of a triangle [closed]

Triangle $ABC$ has all sides of integral length. Vertex $A$ is at $(0,0)$, $B$ lies on the line joining $(0,0)$ and $(3,6)$ and $C$ lies on the line joining $(0,0)$ and $(2,-1)$. Two of the three ...
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### Finding the length of a side of an equilateral triangle

There is a large right isosceles triangle with a hypotenuse length of $24$. Inside the triangle is an equilateral triangle with a vertex on the midpoint of the hypotenuse. If the length of each side ...
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### Solving for length of an unknown side of a triangle.

I have been given the figure below: Figure (click me). I know that $AD=20-x$ and $m\angle ACD=m\angle BCD$. How can I set up a ratio also knowing that $AC=11$ and $BC=14$ in order to find $x$? ...