For questions about triangles

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1answer
37 views

Crazy rectangles, semi-circles, and circles!

Problem is to find the ratio of the area of the circle to that of the semi-circle. Note that points $F$ and $E$ weren't given in the original diagram, and that the circle at the top-right ...
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1answer
18 views

Proving congruency of triangles

Question: Given $AB$ is diameter, $C$ and $D$ lie on circumference, $AB = 15cm$, $AC = 12cm$, $BD = 9cm$, find area of quadrilateral ABCD. Note that the points $O$ and $Q$ were not in the ...
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1answer
28 views

How can solve a triangle knowing its area , one side and an opposite angle

I need to calculate the missing elements of a triangle knowing its area one side and the angle opposite the given side. The triangle is not a right angle triangle, nor is it equilateral or isosceles ...
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2answers
39 views

Side of triangle problem

In triangle $ABC$, $AB=BC=12$. Side $AC$ extended through $C$ a length equal to itself to a point $D$. Point $E$ is on $AB$; $DE$ intersects $BC$ at $F$ and $BF$ equal to 8. Find $AE$ without using ...
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0answers
18 views

Volume of a Part of a Triangular Prism Enclosed in a Sphere

I'm having trouble finding the volume of the shaded prism. I know how to calculate the volume by extending the height of this prism to create a triangular based pyramid, but I cannot get the same ...
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1answer
29 views

Show that the area of the triangle ABC is maximized when $\angle BCA$ = $\angle CAB$

Let A, B, and C be three points on a circle of radius 1. Suppose that the magnitude of $\angle ABC$ is fixed. Then show that the area of the triangle ABC is maximized when $\angle BCA$ = $\angle ...
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1answer
76 views

Finding the missing coordinate of a point within a 3D triangle

We have an equilateral triangle $ABC$ in 3-dimensional space. The points are known, such as: $A = (x_1,y_1,z_1)$ $B = (x_2,y_2,z_2)$ $C = (x_3,y_3,z_3)$ Point $P$ is on triangle $ABC$. If I know ...
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1answer
43 views

$\sqrt{\frac{15}4+\sum\cos(A-B)}\ge\sum\sin A$ in a triangle?

How can I prove that ( $\small{\sum}$ denotes cyclic sum here), for any triangle $ABC$: $$\sqrt{\frac{15}4+\sum\cos(A-B)}\ge\sum\sin A$$ I don't see where to begin even. Any hints would be ...
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0answers
17 views

How to find the length of the union of Isosceles triangles

I am given N number of right angles triangles all of which are also Isosceles triangles. For each triangle, I am told where they start on a number line and where they end on a number line with end ...
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2answers
31 views

Nature of the $\triangle$

In $\triangle$ ABC, the $\angle BAC$ is a root of the equation $3^{1\over2} \cos x + \sin x = {1\over2}.$ Then what kind of triangle is the $\triangle$ ABC.
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1answer
79 views

Circumcircle of an isosceles triangle and length relation

I was asked to prove the following problem. Consider the following diagram where a triangle $ABC$ lies inside its circumcircle, $D$ is the point where the angle bisector $\alpha$ of $B$ intersects ...
3
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1answer
58 views

How to prove $\cos(\frac{B-C}2)\ge \sqrt{\frac{2r}{R}}$?

For any triangle $ABC$, prove that: $$\cos(\frac{B-C}2)\ge \sqrt{\frac{2r}{R}}$$ I have tried many approaches but none seems to work. I noted that $\cos(\frac{B-C}2)=\frac{AM}{2R}$, where $M$ is ...
2
votes
1answer
17 views

Inequality relating to product of sides of convex quadrilateral

We have been that length of both the diagonals are equal to $x$. What can be the maximum value of the product of length of sides? It is obvious that an upper bound exists, but I can't get the ...
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0answers
24 views

Proof metric space with distance function

Thats the first time i have to do such an proof but don't know how, never seen or done this before. Especially (iii). Let $X$ be the Set of all complex sequences. $$ d((a_n),(b_n)) := ...
0
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1answer
20 views

calculate the angles of a triangle?

If we have a triangle $ABC$ and we only know three things: -The angle $A$ -The length $AB$ -The length $AC$ Is it possible to calculate the other angles: $B$, $C$ All what I can think of ...
3
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1answer
77 views

Proving a tough geometrical inequality, with equality in equilateral triangles.

For any triangle with sides $a ,b, c$ prove or disprove (1) and (2) : $$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}\ge \frac34$$ Equality in (1) holds if and only if the triangle is ...
0
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2answers
42 views

No. of equilateral triangles required to completely fill a bigger equilateral triangle

$\triangle ABC$ is equilateral with side length=2.1cm Smaller equilateral triangles with side length=1cm are placed over $\triangle ABC$ so that it is fully covered. Find the minimum number of such ...
0
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1answer
65 views

Formula to calculate a side of triangle with given angle

I have triangle like in the picture. The known angles: α (total angle of the I-J-K2 triangle) b (total angle of the I-P2-K2 and I-P1-K2 triangles) The known 3D points with X,Y,Z-coordinates: ...
3
votes
1answer
44 views

Congruency in bow-tie triangles

We've just started congruency in my class, and we've stumbled across a question which goes like this: Prove that ∆AOB $\equiv$ ∆COD My teacher told us that the way to solve this is using the ...
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2answers
202 views

A geometric inequality, proving $8r+2R\le AM_1+BM_2+CM_3\le 6R$

Here, $AM_1$ is the angle bisector of $\angle A$ extended to the circumcircle and so on. $R$ is the circumradius and $r$ is the inradius, respectively. I have to prove that: $$8r+2R\le ...
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2answers
40 views

Construct an equilateral triangle given a line segment

Given two vertices that make up a line segment (x1,y1) and (x2,y2), how can we find the third vertice that would make up an equilateral triangle? I'm looking to derive the third vertex algebraically, ...
2
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2answers
49 views

Equilateral triange, sum…

Just a short question: In a triangle we have $\sum (\frac{a}{b+c})^24. Is the triangle equilateral? I have derived
3
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2answers
228 views

What is the history of the use of the term “scalene triangle”?

A "scalene triangle" is a triangle with three unequal sides. As far as I can tell, this term is not in much use in serious mathematics — in fact, before I became a high school math teacher, I'd ...
0
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1answer
21 views

How to determine the range of a angle measure?

In $\Delta$ $KLM$, $KL=20$ $LM=13$ m$\angle K$$=40$. What is the range for angle $M$'s measure? Something like between $90^{\circ}$ and $180^{\circ}$
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1answer
42 views

Year 10 - Trigonometry

Please ignore the pencilled 4m in the diagram but I really need to know what the length of the bottom line - line DC - is. A procedure or tips on how to calculate this would be useful. Also, is the ...
5
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1answer
122 views

If this relation holds, then is the triangle equilateral?

Let $ABC$ be a triangle. If $$\sum_{cyc}\frac{BC}{4AC\cos^2({\frac{\angle BAC}{2})}+BC}=\frac{3}{4}$$ then the triangle is equilateral? We can check if we set $\widehat{BAC}=\pi/3$ and $AB=BC=CA$ that ...
0
votes
1answer
16 views

New Angle When Opposite Side is Halved

Suppose you have a right triangle with any length sides. The value of one of the angles is $\theta$ and the opposite side is a. If I change the triangle so that the new length of side a is $\frac a2$, ...
2
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0answers
41 views

Prove that the maximum volume of a triangular-base prism is $\sqrt{\dfrac{K^3}{54}}$ where K is the area of three triangles containing a vertex A

Consider a prism with triangular base. The total area of the three faces containing a particular vertex $A$ is $K$. Show that the maximum possible volume of the prism is $\sqrt{\frac{K^3}{54}}$ and ...
3
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1answer
106 views

Inequality problem about sides of a triangle and the semiperimeter

Let $a,b,c$ the sides of a triangle and $s$ be the semi perimeter. Then show that $$ a^2+b^2+c^2 > \frac{36}{35}(a^2+\frac{abc}{s}) $$ I tried it doing in many ways using some ...
1
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1answer
27 views

Find a right angle triangle in with 3 vertices and one parameter

Given three coordinates, which could be $A=(7,3)$, $B=(2,4)$, $C=(k,-2)$ I want to find the values of $k$ that make a right angle diagram out of the three points. So I initially was thinking to find ...
0
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1answer
35 views

An Inverse Cosine Problem

Here is my problem: $$ \sin(\cos^{-1} \frac{2}{5} ) $$ I know how to do it for the most part; I just draw a triangle with sides 2,5 and √21 and I then find the sine (opposite/hypotenuse) of the ...
1
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1answer
52 views

Finding the largest angle of a triangle

The sides of a triangle are $(x^2+x+1), (2x+1)$ and $(x^2-1)$. Then what is the largest of the 3 angles of triangle?
1
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1answer
32 views

Sin and Cos relationship with Triangle sides

In a triangle ABC, ${sinA < \frac{a}{c}}$ and ${cosA > \frac{b}{c}}$. Which of the statements below are always false regarding triangle ABC? ABC is an acute triangle ABC is an isosceles ...
0
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1answer
53 views

Squares constructed externally on the sides of a triangle and concurrent lines

On the sides $BC, CA$ and $AB$ of the triangle $ABC$ we construct externally the squares $BCDE, ACFG $ and $ABHI$. Denote $A', B'$ and $C'$ the intersectiond points of the lines $BF$ and $CH$, $AD$ ...
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5answers
2k views

Tricky Triangle Area Problem

This was from a recent math competition that I was in. So, a triangle has sides $2$ , $5$, and $\sqrt{33}$. How can I derive the area? I can't use a calculator, and (the form of) Heron's formula (that ...
1
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2answers
25 views

What is the nature of Triangle if AB/AC=1/2 angle (BAC)=60°

What is the nature of Triangle if $\frac{AB}{AC}=\frac12$ and $\angle BAC=60^{\circ}$?. Can we use ratio between side lengths?
1
vote
1answer
43 views

Bisectors and equilateral triangle

I have the following problem: bisectors $AA',BB'$ and $CC'$ of the triangle $ABC$ intersect the circumcircle in the points $A",B",C".$ Holds the following equivalence: ...
1
vote
1answer
25 views

Why is the length of one of the segments negative in the Menelaus' theorem? Aren't all distances by definition positive?

Why is the length of one of the segments negative in the Menelaus' theorem? Aren't all distances by definition positive? I think we all know the Menelaus' theorem, which claims that the following ...
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votes
3answers
64 views

Equilateral triangle inscribed in a ellipse

"Given any point on a ellipse, is it always possible to inscribe an equilateral triangle, with a vertex coincident with that point, in the ellipse?" I thought I could use analytical geometry, but ...
3
votes
1answer
51 views

Is there any quantity related to $\cos \left(\frac{B-C}{2}\right) + \cos \left(\frac{C-A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$

While doing an inequality, I encountered the following expression,where $ABC$ is a triangle: $$\cos \left(\frac{B-C}{2}\right) + \cos \left(\frac{C-A}{2}\right) + \cos \left(\frac{A-B}{2}\right)$$ ...
0
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1answer
19 views

Using Right triangles to determine Values

Missed a day of class, and I can't seem to figure out the concept here. It seems simple but I just can't wrap my head around it. Any and all help is much appreciated.
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1answer
30 views

Question that includes Trigonometry

In the diagram, $AB = 80 cm$, $\angle ABD = 44^∘$ (Angle B), $\angle BAC = 31^∘$, $\angle DAC =37^∘$ and $\angle DBC = 36^∘$. Calculate: a) $BC$ b) $BD$ c) $CD$
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votes
2answers
54 views

Formula to calculate a length to a point on hypotenuse according to given angle

I have a right triangle: Height: y (value over 0) Width: y (value over 0) Angle: α (degrees, value between 0-90) I need to find out the formula to count the length of x.
0
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1answer
18 views

Sum of edge numbers for triangle given starting number, increment and number of levels

For example, if starting number (N) = 1, increment (I) = 5, and number of levels (L) = 4, you get the following triangle: 16 11 11 6 6 1 1 ...
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3answers
33 views

Is this some kind of triangle inquality?

I stumbled upon the following inequality: $$\Vert x+hz-(x+y)-(p-(x+y))\Vert_2 \geq \Vert p-(x+y)\Vert_2-\Vert x+hz-(x+y)\Vert_2$$ where $p,x,y,z \in \mathbb{R}^n$. My question is: Is this some kind ...
5
votes
2answers
141 views

Relationship between circles touching incircle

I am trying to derive a relation between radius of those outer circles and radius of the incircle. Those outer circles are tangent to the incircle and respective sides. I have tried and failed ...
0
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1answer
31 views

Altitudes of Triangle

I have a triangle defined as 3 lines, each defined by two coordinate points A and B. I have the area of the triangle but need to calculate the 3 altitudes and their respective sides A and B points. ...
2
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3answers
32 views

Trignometric Question [closed]

The elevation of the top of a tower $KT$ from a point $A$ is $27^\circ$. At another point $B$, $50$ meters nearer to the foot of the tower where $ABK$ is a straight line, the angle of elevation is ...
0
votes
1answer
16 views

Find term for one angle of two in a trig function

In a right angled triangle, I know that $\tan (x) = \cfrac{4}{z}$ and that $\tan(x+y) = \cfrac{12}{z}$. I need to find an equation which has only $\tan(y)$. The answer is $\cfrac{12}{z} = ...
2
votes
0answers
105 views

Proving there is no set of five distinct points s.t. every three points are the vertices of a right triangle.

We can see that the following proposition is true. Proposition : Each triangle $ABD, ACD, BCD$ is a right triangle for $$A(0,b,0), B(a,0,0), C(0,0,0)\ \ \ (a\gt 0, b\gt 0)$$ $\iff D$ is either ...