# Tagged Questions

For questions about properties and applications of triangles

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### What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles?

What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles? My work so far: Case 1) let $ABC -$ an acute-angled triangle. Then radii of the ...
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### How to solve this problem about the reciprocals of the angles inside an acute triangle?

I was solving some previous year papers and stumbled upon this question. I don't even know how to approach this problem. The problem is as follows: Consider an acute angled triangle $PQR$ such ...
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### Ratio of areas in triangles with Cevians

Can someone show me how to solve this problem as they normally would? It would also be very helpful if you could also give me a mass points solution. I know mass points works for ratios of segments, ...
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### Geometry and property of triangle

$ABC$ is a cyclic triangle and bisector of angle $B\widehat{A}C$, $A\widehat{B}C$ and $A\widehat{C}B$ touches circle at $P$, $Q$ and $R$ respectively then measure of angle $R\widehat{Q}P$ is? The ...
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### Cover a polygon with polygons

Besides right angled triangles, is there any polygon I could use to cover any given (regular or not) polygon? It's clear that given a triangle, square, hexagon or rectangle you would other options. ...
1answer
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### How to do Vectors with triangles?

This is my homework and I have just started learning it and I don't really quite understand it.
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### Sampling from Irwin-Hall distribution using triangular distribution

So I need to sample from the Irwin-Hall distribution using rejection sampling with the triangular distribution. I built 2 functions: The first is d_irwin which receives an $x\in supp(g)$ and the n we ...
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### Calculate the coordinates of two points in an isosceles triangle

In the triangle below, given the point A, angle θ and length d of the two equal sides, how can the points B and C be calculated? Edit:After brainstorming for quite some time, I ended up with a ...
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### Relationship between incenter and circumcenter

Let ABC be an acute triangle with circumcenter O and incenter I. Points E, M lie on AC and F, N on AB so that BE ⊥ AC, CF ⊥ AB, ∠ABM = ∠CBM and ∠ACN = ∠BCN. Prove that I lies on EF if and only if O ...
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### Find area of triangle ABC given areas of sub-triangles

The line p is parallel to the the side AB of triangle ABC and splits the sides AC and BC in points D and E, respectively. If the area of triangle ABD is m and the area of triangle AEC is n, find the ...
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### Finding sides of triangle

Given : $$\triangle ABC$$ $$M \in AB,N \in BC ,P \in AC$$ are the points at which the incircle crosses the triangle $$MN=3\sqrt{10}$$ $$NP=2\sqrt{20}$$ $$PM=10$$ I have to find the sides of the ...
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### Taylor's formula and its quadratic term

I struggle with the following problem: For a function $$f: \mathbb{C} \rightarrow \mathbb{R}~,$$ $f$ attains its maximum for $z_0= e^{i\pi/3}$, $f(z_0)=F_{max}.$ Assume we may use Taylor's theorem (...
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### Circumcentre and Incentre [closed]

if I is the incentre and S is the circumcenter of ABC prove that angle IAS is half the difference between angle B and angle C
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### Circle inscribed in Equilateral Triangles

The circle inscribed in the triangle $ABC$ touches the sides $BC$ , $CA$ , and $AB$ in the points $A_1,B_1,C_1$ respectively. Similarly the circle inscribed in the triangle $A_1B_1C_1$ touches the ...
1answer
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### Point which minimizes the squared sum distances to edges of a triangle [duplicate]

I need to find coordinates of a point at which the sum of the squared distances to each of the three lines that form a triangle is minimized. It seemed to me that the point is the triangle's incenter,...
1answer
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### X is any point on AB and the median AD of triangle ABC meets XC at Y.Prove that XY/YC = AX/XB

X is any point on AB and the median AD of triangle ABC meets XC at Y.Prove that XY/YC = AX/XB.
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### Write down the iterated integral which expresses the surface area of z = y 6 cos3 x over the triangle with vertices (-1,1), (1,1), (0,2):

This problem has been done a few times on other sites but there is no work or explanation of the steps taken to get h(x,y). I can understand getting the limits by finding the bounds of the triangle so ...
1answer
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### Finding angle associated with point inside an equilateral triangle.

$\triangle{ABC}$ is an equilateral triangle. $|AD|=6$. $|BD|=10$. $|CD|=8$. What is $m\angle{CDA}$? First thing comes to mind is Ceva theorem. I used its trigonometric form to reach ...
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### triangle park problem [closed]

We have a park that is triangle. We don't know the shape of the triangle and it can have any triangle shape and lengths. Where should we place a lamp to have light everywhere in the park? my english ...
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### Pythagorean Theorem on Spiral of Theodorus Triangles

I have 1 right triangle of dimensions $\sqrt75$$, 11, 14$. I'd like to know how to quickly obtain the other right triangles with $\sqrt75$ as a leg, and two integers as the hypotenuse and the other ...
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### How to calculate triangle coordinates in cartesian plane?

My problem can be describe by following image: I know coordinates of an example P point. Say, they are equal to (8,8). I also ...
4answers
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### Properties of Equilateral Triangles in Circles

If there is an equilateral triangle in a circle, would the midpoint of any of the 3 sides be half the radius? e.g if the radius was 6 and at the midpoint of the triangle (call it B) would center to ...
1answer
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### Proof formula for triangle: $b^2|BD| +a^2|AD| - c|CD|^2 = c|AD||BD|$

given is a triangle with corners A,B,C and corresponding sides a, b, c. D is a point somewhere between A and B. I have to proof: $b^2|BD| +a^2|AD| - c|CD|^2 = c|AD||BD|$ Unfortunately I have no ...
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### Find the width/height of a triangle given a side length and two lines

I'm a programmer and I came across a math problem in my current project that I can't figure out. My situation looks like this. Everything in black is known or I know how to figure out. ...
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### Probability that one part of a randomly cut equilateral triangle covers the other without flipping

At Probability that one part of a randomly cut equilateral triangle covers the other, the case with flipping allowed was quickly solved. The case without flipping seems more difficult and hasn't been ...
1answer
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### Probability that one part of a randomly cut equilateral triangle covers the other

If you make a straight cut through a square, one part can always be made to cover the other. (This is true by symmetry if the cut goes through the centre, and if it doesn't, you can shift it to the ...
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### Division of Solid Angle When Subdividing Spherical Triangle

Suppose I have a spherical triangle (no special properties; in particular, not equilateral) with a known solid angle. Now, I divide it into four new spherical triangles by bisecting each edge: ...
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### Proving the Pythagorean Theorem with just variables

I basically have just three problems: a) How many similar triangles can you find in the figure below? b) Use part a) to prove the Pythagorean Theorem (note: to prove something you do need to ...