For questions about properties and applications of triangles

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Triangle inscribed in another triangle

If $a,b,c$ are the sides of a triangle,$\lambda a,\lambda b,\lambda c$ the sides of a similar triangle inscribed in the former and $\theta$ the angle between the sides $a$ and $\lambda c$,prove that ...
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1answer
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orthocentre and triangle related question

$AD$, $BE$, and $CF$ are the altitudes of triangle $ABC$ with orthocentre $H$, then $C$ is the orthocentre of which triangle? Answer: triangle $ABH$. Please explain.
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2answers
54 views

Does the centroid of a triangle ever fall outside of its Morley's triangle?

Let $T$ be a triangle, and $M$ its (first) Morley triangle:                     (Image from Bruce Shawyer web page.) Q1. Does the ...
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1answer
84 views

Prove that $\tan\alpha =\tan^{2}\frac{A}{2}.\tan\frac{B-C}{2}$

Given a triangle ABC with the sides $AB < AC$ and $AM, AD$ respectively median and bisector of angle $A$. Let $\angle MAD = \alpha$. Prove that $$\tan\alpha =\tan^{2}\frac{A}{2}\cdot ...
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1answer
61 views

Point on the Plane, a Triangle, and a Lower Bound of a Ratio Sum

Let $ABC$ be a triangle on the Euclidean plane. At which point $P$ on the plane does the ratio sum $\frac{PA}{BC}+\frac{PB}{CA}+\frac{PC}{AB}$ attain its minimum value? Prove also that, for any ...
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2answers
49 views

Interesting problem in congruence of triangles

While solving the exercises of my book I came across this interesting problem: $\triangle ABC$ is isosceles triangle with $AB=AC$. D is a point on base BC such that $AD$ perpendicular on $BC$. To ...
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Given the perimeter of an equilateral triangle, and a fraction of the perimeter, how would i find the (x,y) coordinates of the triangle.

Given the perimeter of an equilateral triangle, and a fraction of the perimeter, how would I: Create a formula for finding the x coordinate Create a formula for finding the y coordinate of the ...
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6answers
560 views

Wanted : for more formulas to find the area of a triangle?

I know some formulas to find a triangle's area, like the ones below. Is there any reference containing most triangle area formulas? If you know more, please add them as an answer ...
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3answers
28 views

Scalene triangle with semicircles mensuration

I was recently going through a mensuration sum from a tenth grade board exam book. This one particular question stumped me, and I spent the entire evening thinking of this, but to no avail. The ...
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26 views

Is my proof of 'inscribed angle theorem' different from the usual one?

The usual proof of inscribed circle theorem uses the fact that the sum of interior angles is equal to exterior angle. I've formulated a little different approach, although the underlying concept is ...
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2answers
59 views

Substitute for finding the hypotenuse of a right triangle?

All of us know the way to calculate the hypotenuse of a right triangle: Using the Pythagorean Theorem. I came up with a substitute to this. Let the shortest leg of the right triangle be '$a$' units, ...
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2answers
47 views

What is the symbol to denote that two triangles are similar?

Does there exist a unique symbol to denote that two triangles are similar to each other without resorting to using the phrase "is similar"?
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36 views

What can be said about triangle with certain condition?

This question comes from 1988 Irish Mathematical Olympiad, for all those interested. A mathematical moron is given the values $b,c,\alpha$ for a triangle $ABC$ and is required to find $a$. He does ...
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0answers
24 views

Given a single point in 3d space, and 3 points that make up a triangle, find the closest point in/on the triangle to the point.

Given point $(p,q,r)$ and 3 points which make up a triangle, find the closest point in the triangle to the point in space. From the triangle, we can find the equation of the plane $Ax+By+Cz+d=0.$ ...
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0answers
28 views

Finding maximum subset-triangles

To a given base (ab), triangles are constructed by choosing a point (p). How can i find the maximum subset-triangles(*)? (*)subset-triangle: p' is inside the triangle abp. Allowed interceptions from ...
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0answers
22 views

What is the relationship of these numbers?

I have two problems that closely relate to each other. I am working with angles. When the angle of Y is 90 degrees the answer to the first problem 360 degrees, while answer to the second is 180 ...
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2answers
32 views

Find a bisector point of a circle

The coordinates of $A=(x_{0},y_{0}$) and $B=(x_{1},y_{1}$) are given. How to find the coordinates of $C$ and $D$ as per given information below. ABC is equilateral triangle such that $AB=BC=CA$ ...
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22 views

Area of equilateral triangle from circumcircle

I am trying to calculate skewness of triangle. Given the sides of a triangle (not equilateral), I calculated circumradius from which I would like to get area of equilateral triangle.
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1answer
25 views

What are the unknown lengths are angles for the roof?

Sharma is building a shed and wants to determine the measurements for the roof. The span of the roof will be 10 feet and she plans to use a 5:12 roof pitch. This means that the roof will rise 5 ...
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0answers
475 views

Two circumcircles of triangles defined relative to a fixed acute triangle are tangent to each other (IMO 2015)

I'm posting here the question because I want to see a nice synthetic solution (not using complex numbers or inversive geometry) for the 3rd problem from IMO 2015. The problem is as follows: Let ...
3
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1answer
70 views

Why can't the nth triangular number be expressed as the area of an equilateral triangle?

It should be self-intuitive that the $nth$ triangular number is an equilateral triangle with base $n$, and thus its area should equal the value of the triangular number. So, I was wondering: why ...
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1answer
56 views

How to know which side of the right angled triangle is the base?

If we are given a right angled triangle without any angle or length of any side. How we will find that which side is the base, which side is the perpendicular.
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2answers
35 views

Triangle and Ratio : Find the length of a side.

Let $\theta = \angle CAD, \phi = \angle CDB, \varphi=\angle DBC, \alpha = \angle BCD$ and $\beta=\angle ACD$. Then we have the following system of equations $\theta + \varphi = 90^{\circ},$ ...
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3answers
51 views

Prove $\triangle{ABC}$ is isosceles if $\cos A = \frac{\sin B}{2\sin C}$

In $\triangle{ABC}$, $$\cos A = \frac{\sin B}{2\sin C}$$ How to prove that $\triangle{ABC}$ is isosceles?
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At what angle does the stone need to be hit?

In curling, it is often necessary to hit and displace an opponent’s stone to win the end. Olivia would like to hit her opponent’s stone with her own stone. If she releases her stone at the hog line, ...
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5answers
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Pythagorean theorem expressed without roots in an old Tamilian (Indian) statement

There's an old Tamil statement that predicts the hypotenuse of a right angle triangle to a reasonable level of accuracy considering it doesn't involve roots. This is how it goes: “Odum Neelam ...
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2answers
59 views

Find the Vertices of a Triangle from Set of Points

I have a set of cartesian x,y points which I know am fairly certain are on the edges of a triangle. What is the easiest way (either algebraically or algorithmically) to identify what the three ...
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2answers
51 views

How do I properly read a clinometer?

If the weight hangs down at roughly 42 degrees, would the angle be 90 degrees - 42 degrees = 48 degrees?
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3answers
34 views

What is the unknown angle?

So first off I started with the pythagorean theorem to find the missing leg of the triangle. \begin{align*} 5^2 + b^2 ={}& 8^2 \\ 25 + b^2 ={}& 64 \\ 64 - 25 ={}& 39 \\ \text{missing ...
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2answers
101 views

Show that in any triangle, we have $\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$

Show that in any triangle, we have $$\frac{a\sin A+b\sin B+c\sin C}{a\cos A+b\cos B+c\cos C}=R\left(\frac{a^2+b^2+c^2}{abc}\right),$$ where $R$ is the circumradius of the triangle. Here is my work: ...
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The inequality $\frac{MA}{BC}+\frac{MB}{CA}+\frac{MC}{AB}\geq \sqrt{3}$

Given ∆$ABC$ and $M$ is an interior point. Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$ When does equality holds?
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3answers
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Find the area of triangle, given an angle and the length of the segments cut by the projection of the incenter on the opposite side.

In a triangle $ABC$, one of the angles (say $\widehat{C}$) equals $60^\circ$. Given that the incircle touches the opposite side ($AB$) in a point that splits it in two segments having length $a$ ...
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1answer
24 views

What is the angle of b?

So first off, I know how to find the missing length of the leg of the triangle using the pythagorean theorem. $6^2 + b^2 = c^2$ $36 + b^2 = 100$ $100 - 36 = 64$ $\sqrt{64} = 8$. So angle angle ...
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2answers
43 views

What angle does the board need to be cut at?

If someone has a 2'' wide board and a 1 1/2'' wide board, and they want to cut the narrower board at an angle so the cut is 2'' long and the boards will fit together, what angle do they need to cut ...
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2answers
45 views

If a 16' ladder is placed correctly on a level surface, how high up will the ladder reach?

So i have just began learning about sin cos and tan, and i came across this problem and for some reason I'm having trouble figuring it out. *** When using a straight ladder, it is recommended that ...
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1answer
45 views

Explain why two right triangles, each with an acute angle of 17 degrees, must be similar.

Two right angles with an acute angle of 17 degrees must be similar because triangles that are similar share the same angles.Is this proper?
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4answers
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The position of a ladder leaning against a wall and touching a box under it

I was reading a newspaper and there was a little math riddle, I thought "how funny, that's gonna be easy, let's do it" and here am I... The problem goes as follow : in a barn, there is a 1 meter ...
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1answer
41 views

Find angles between sides of triangle and coordinate planes ($xy,yz,zx$ planes) using three 3d vectors .

Given the following, three vectors: \begin{align*} \vec{a}& = 3i−2j+5k, \\ \vec{b}& =i−6j+6k, \\ \vec{c}& =2i+3j−k, \\ \end{align*} find the angles between sides of triangle and ...
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3answers
49 views

Is it possible to find the vertices of an equilateral triangle given its center point?

I was wondering how to find the vertices of an equilateral triangle given its center point? Such as: ...
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2answers
69 views

Find perimeter and angle of triangle using three 3d vectors .

Given the following, three vectors: $$\vec{a} = 3\mathrm{i} - 2\mathrm{j} + 5\mathrm{k}\\\vec{b} = \mathrm{i} - 6\mathrm{j} + 6\mathrm{k}\\\vec{c} = 2\mathrm{i} + 3\mathrm{j} - \mathrm{k},\\$$ find ...
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2answers
71 views

Minimum value of cosA+cosB+cosC in a triangle ABC

I have used jensen's inequality but couldn't move on.
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2answers
43 views

How would you find the length of a side of a triangle where 2 sides are known and the length of a line in the middle is also known?

How would you find the length of a side of a triangle where the other 2 side lengths are known and the length of a another line that meets at the same point is known? I know there has to be an answer ...
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1answer
58 views

Proving that $ ABC$ is similar to $DQP$

Let $G$ be the centroid of triangle $ABC$. Let $D$ be the midpoint of $BC$. A line through $G$ parallel to $BC$ meet $AB$ at $M$ and $AC$ at $N$. $MC$ meets $BG$ at $P$ and $NB$ meets $CG$ at $Q$. ...
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Alternative proof for the equality of two angles in an isosceles triangle.

From the answers of my previous question, I got an idea to prove equality of two angles in an isosceles triangle. In that question the equality of two angles in a right-angled-isosceles triangle was ...
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0answers
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Generalization to higher dimensions of a statement about plane triangles

Let $\Delta=\Delta ABC$ be a plane triangle with area $F_\Delta$ and let $P$ be a point in $\Delta$. Draw lines through $P$ parallel to the sides of $\Delta$; then $\Delta$ is decomposed into three ...
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3answers
88 views

How is $\sin 45^\circ=\frac{1}{\sqrt 2}$?

I've been reading about the proof of $\sin 45^\circ=\dfrac{1}{\sqrt 2}$ in my book. They did it as following, let $\triangle ABC$ be an isosceles triangle as shown, Since the triangle is isosceles ...
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2answers
42 views

Elementary problem in geometry [closed]

The problem asks to find the angle at $C$. The distance between $A$ and $B$ is $12 \space m$ and the distance between $B$ and $C$ is $8\space m$. Anyone got an idea?
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1answer
27 views

Question related to triangles.

I am stuck at a question: O is a point in the interior of ∆PQR , then which of the following is true: 1)$(OP+OQ+OR)<1/2(PQ+QR+PR)$ 2)$(OP+OQ+OR)=1/2(PQ+QR+PR)$ 3)$(OP+OQ+OR)>1/2(PQ+QR+PR)$ ...
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2answers
42 views

Special triangles

I have this question that I have the answer to but no working how to get it, is it by pure memorization of angles or there some steps? Without a calculator, determine, in radians, the angles of a ...
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4answers
35 views

Prove that the co-ordinates of the centroid of a triangle is an average of that of vertices

For a given triangle [ABC], how do I prove that the co-ordinates of the Centroid $O_{xy}$ (intersection of the medians) is the average of the individual vertices? $O_x = \left(\frac {A_x + B_x + ...