For questions about properties and applications of triangles

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1answer
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liquid triangles

note: for simplicity the 2-dimensional case is described here. a similar situation could be treated in higher dimensions. liquids are distinguished by their ability to change form whilst retaining ...
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2answers
53 views

Relation between the radius and the area of tangential polygon

I've recently found a book with loads of formulas for triangle area, but unfortunaly the formulas were just listed, there wasn't a proof for them. I've tried to proof them. But I've stopped at one of ...
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5answers
284 views

Mensuration question

I recently came across a puzzling question: Two rectangles ABCD and DBEF are as shown in the figure. The area of DBEF is: Figure (hand-made): I know that through Pythagoras, we get ...
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1answer
464 views

Maximum area of a rectangle inside a triangle

I recently came across a problem where it gave a triangle with integer side lengths, and it asked you to find the maximum area of a rectangle of a triangle. I solved the problem correctly, but it ...
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1answer
78 views

Angle bisector in a triangle

For the angle bisector $I_a$ in a triangle $ABC$ it holds $$I_a^2 = \frac{bc}{(b+c)^2}[(b+c)^2 - a^2]$$ If $I$ is the incenter, I wonder if there exist similar formula for the part $AI^2$.
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2answers
849 views

how many rectangles in this shape

I've learned in my high school the solution to such riddle: How many rectangles are there in this shape: the solution is through combinations: in this shape is a $5\times 6$ grid so the number of ...
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1answer
121 views

Triple systems with no six points carrying three triangles

Can anyone please send a link to this article? ...
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2answers
215 views

Find value of the angle x

Find the value of the angle x. Plus : Someone could recommend me some good book about this subject ?
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1answer
63 views

Formula in a triangle

Let $H$ be the orthocenter in a triangle with sides $a, b, c$. Is it true that $$a^2 + HA^2 = 4R^2$$ where $R$ is the circumradius?
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1answer
99 views

Inequality in a triangle

Let $O$ be the circumcenter and $H$ the orthocenter in a triangle with sides $a, b, c$. Is it true that $$aOA^2+bOB^2+cOC^2 \ge aHA^2 + bHB^2 + cHC^2$$ or equivalently $$(a+b+c)R^2 \ge aHA^2 + bHB^2 + ...
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2answers
152 views

How prove this stronger than Weitzenbock's inequality:$(ab+bc+ac)(a+b+c)^2\ge 12\sqrt{3}\cdot S\cdot(a^2+b^2+c^2)$

In $\Delta ABC$,$$AB=c,BC=a,AC=b,S_{ABC}=S$$ show that $$(ab+bc+ac)(a+b+c)^2\ge 12\sqrt{3}\cdot S\cdot(a^2+b^2+c^2)$$ I know this Weitzenböck's_inequality $$a^2+b^2+c^2\ge 4\sqrt{3}S$$ But my ...
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3answers
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Construct triangle given inradius and circumradius

If we know the inradius $r$ of a triangle and the circumradius $R$ we can find out the distance between the incircle $I$ and the circumcircle $O$: $OI^2 = R^2-2Rr$. Therefore we can draw the incircle ...
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1answer
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A Pythagorean problem

We have two points F1, F2. F1-F2 is 21m. We have a point (P) outside the line. The line from F1-P is called D1. The line from F2-P is called D2. P is 12m away from F1-F2 on a straight line crossing ...
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2answers
124 views

Area of an equilateral triangle divided by three lines

An equilateral triangle is divided by three straight lines into seven regions whose areas are shown in the image below. Find the area of the triangle. How to solve this problem ?
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1answer
77 views

construct an equilateral triangle with out knowing its scale

How do i construct an arbitrary equilateral triangle with out knowing its scale? for e.g. pick two points a and b. make $60$ degree acute angles at point $a$ and point $b$ and the two angles meet at ...
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4answers
106 views

How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter?

This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and ...
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1answer
62 views

Prove: Exactly a quarter of 3-part partitions of numbers >2 equal to 0, 2, 10 mod 12 will make a triangle.

Consider perimeters $>2$ equal to $0$, $2$, or $10 \mod(12)$. The sequence starts $10, 12, 14, 22, 24, 26, 34, 36, 38, 46, 48, ...$ and we can look at the three part partitions that make triangles. ...
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1answer
75 views

Which theorem should be used to solve this question?

My friend sent me that question and said "another Carnot theorem" is used to solve this question but i couldnt find that theorem. Can you help me? Additional explanation: $$ \widehat{ABD} = ...
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3answers
339 views

If two sides of a triangle are equal, and the angle between them is $60^\circ$, prove the third side is equal to the first two sides.

In other words, given points $A$ and $X$. Rotate $X$ $\,-60^\circ$ around $A$ to get point $X'$. How would you prove $XX' = AX = AX'$? I know this is true.
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1answer
91 views

explaining the resriction $b<a<2b$ in a triangle

I saw in a book that if $ABC$ is an isosceles triangle $(AB=AC)$ and the triangle is tangent to a circle in points $D,C$ and $AC$ is intersecting the circle in point $E$; $AC=a$, $BC=b$ so it has ...
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1answer
30 views

Figuring out the side of a triangle

I'm having trouble on this problem I don't know how to set it up. I know XO=2 and OB=6. I'd appreciate any hints.
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Triangle inscribed insemicircle area-ratio question

My approach: $m<A= 60$ degrees and $m<C=30$. This creats a 30, 60, 90 triangle with ratios $$1:\sqrt3:2$$ After getting the ratio's of the areas, I obtain $$\frac{b*h}{\pi r^2}=\frac{1*\sqrt ...
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1answer
58 views

height and bisector in a right triangle

We have right angle $ABC$ where $AB$ is hypotenuse, and angle bisector $CD$ we know that $AB=c$, $CD=u$. Express height depending on the $c, u$, and what is the condition between $c,u$ that the ...
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2answers
64 views

Number of Equilateral triangles in circle with 42 evenly spaced points?

I know that the answer is 42/3 = 14 points, or in general for a circle with N points it is N/3, but I don't know why it actually works. Why is the number of equilateral triangles for a circle with N ...
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1answer
145 views

Find Cathetus C1, C2 Knowing Hipotenuse or Find C1, C2, C3, C4 of Rectangle

I have a rectangle. I know all sides and 4 points for it (see black rectangle below). I resize one edge of this rectangle to any point (see resized red color rectangle and new point B). Here is the ...
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1answer
45 views

Computing a normal to triangle

The normal to surface is just a vector that is prependicular to the surface in particular point, right? So normale to the triangle is a vector perpendicular to his plane? I have a triangle, for ...
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1answer
119 views

Calculating radius of the circle

A perfect circle sits exactly within a right-angled triangle,dividing its hypotenuse into two segments of 3 and 10 units. The area of the triangle is 30 square units. What's the radius of the ...
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1answer
671 views

Is there a reflex angle symbol?

I only knew this angle symbol "$\angle$", which is usually used to represent acute angles. But now I have accounted a problem, where I wanted to represent the reflex symbol of $\angle ABC$. I know, I ...
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1answer
135 views

What is the shape of the region where a special point can exist?

For an equilateral triangle $ABC$ which has edge-length $1$, let $D,E,F$ be a point of contact of the inscribed circle and an edge $BC, CA, AB$ respectively. Then, let us call the region surrounded ...
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1answer
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Heronian isosceles triangles

This is a problem from Project Euler, problem 94. The problem asks about isosceles triangles with integer sides (differing by 1 unit, e.g, 5-5-6) and integer area, which are known to be Heronian ...
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1answer
130 views

Obtuse-angled triangle equation

Give a obtuse-angled triangle and the obtuse angle is 105º. Find n such that the acute angles be the roots of the equation. ...
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3answers
75 views

Area of the triangle a,b,c

How to prove in any triangle that the area $X$ is given by: $$X=\frac{1}{4}(a+b+c)^2\tan \frac{A}{2} \tan \frac{B}{2} \tan \frac{C}{2}$$
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3answers
65 views

Prove that triangle is equilateral

I have a triangle here, how do I prove that $BCD$ is equilateral(so all lines have the same length) And yes this is 2D What I have so far is $$BAC = 120\deg$$ So how do I point out that $$BCD = ...
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1answer
58 views

Length of hypotenuse

Let a circle centered at $O$ have radius $OA=10$. Let OB be perpendicular on OA.Let G and E be points respectively on on OB and OA.Let F be a point on the circumference such that GFEO is a ...
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1answer
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Sufficient condition for graph to have triangle

I need a sufficient condition for graph to have triangle (exists $3$ vertices, each $2$ of them are connected by edge). I think it should be number of edges or the degree for vertices but didn't find ...
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2answers
59 views

Triangle and Maxium value

Given any triangle ABC with $a \ge b \ge c$ such that $\frac{a^3+b^3+c^3}{\sin^3(A)+\sin^3(B)+\sin^3(C)}=7$, what is the maximum value of $a$?
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2answers
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Triangle geometry - synthetic proof

I'm looking for a nice synthetic proof of the following fact. Consider a non-isosceles triangle, pick a vertex. Assume that the median and the altitude passing through this vertex are isogonal ...
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2answers
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A problem of a triangle

Consider a triangle $ABC$ where the median $CM$ is perpendicular to the angle bisector $AL$ and their ratio is $ \sqrt2 : 1 $. The question is to find $\cos A$. Hints? Btw, I do know that the ...
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1answer
111 views

Angle between two rectangles rotated around a point with a gap inbetween

I am trying to find the angle between two rectangles when there is a known gap between them. See this diagram: I have simplified the problem into three triangles, two of which are the same. Here ...
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1answer
80 views

Möbius Transformation of Triangles

I understand that Möbius transformations are angle preserving transformations. Knowing this, my professor asked us to think about how the image of equilateral triangle is not an equilateral triangle ...
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1answer
66 views

Number of ways to form isosceles triangle by picking points on a circle

Given a circle with 24 evenly spaced points, how would you find the number of possible isosceles triangles (which includes equilateral) that can by drawn using the points? My attempt was to say that ...
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1answer
94 views

Prove that $\|a\|+\|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane.

Prove that $\|a\| + \|b\| + \|c\| + \|a+b+c\| \geq \|a+b\| + \|b+c\| + \|c +a\|$ in the plane. Gentle hints only, please! I know that attempting to decompose R.H.S. into $$\alpha a + \beta b + ...
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1answer
597 views

Solving circle's radius only knowing angle & lengths of external triangle OR solving for sides of a triangle partial side lengths

Is this possible? Given that I know the length of Y and Z and the angle of X can I figure out the radius A? If I can't without more information, I can produce another set of data X Y Z at a ...
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3answers
922 views

Minimum distance between point and face

Given a point in 3D space of the form (x, y, z) and a triangle consisting of 3 vectors (also in the (x, y, z) format), how would I calculate the minimum distance between the point and the face of the ...
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0answers
222 views

Question on Proof of Shoelace Formula

I was looking for a way to prove the shoelace formula when I found this proof: For this clockwise order to make sense, you need a point O inside the polygon so that the angles form $OA_{i}A_{i+1}$ ...
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1answer
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$3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle

I am completely lost in this one $3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle, its area is closest to the which integer?
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2answers
208 views

$m$ be the number of distinct non congruent integer sided triangles each with perimeter $15$

Let $m$ be the number of distinct non congruent integer sided triangles each with perimeter $15$ and $n$ be the number of distinct non congruent integer sided triangles each with perimeter $16$ Then ...
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1answer
260 views

Finding the interior angle between two lines of slopes $m_1$ and $m_2$ from a programming perspective

I have been working on a 2-dimensional object creator program that handles manipulations of arbitrary shapes and calculates collision detection between them. The program allows you to input a shape's ...
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1answer
78 views

Where am I wrong in finding area of this triangle?

I was self-reading Mathematics for Economists by Simon and Blume. On page 815, Section 29.4, he has discussed "Norms on Function Space". And here I am stuck: Let $$f_n = \begin{cases} 2n^2-2n^3x, ...
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152 views

Area of triangle $OAB$

Question is : Consider a circle of unit radius centered at $O$ in the plane. let $AB$ be a chord which makes an angle $\theta$ with the tangent to the circle at $A$ .find the area of triangle $OAB$ ...