For questions about properties and applications of triangles

learn more… | top users | synonyms

2
votes
2answers
59 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 ...
0
votes
1answer
39 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 ...
1
vote
1answer
187 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 ...
4
votes
1answer
48 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 ...
1
vote
0answers
46 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 ...
2
votes
2answers
39 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.
3
votes
1answer
259 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 ...
4
votes
1answer
86 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
36 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 ...
0
votes
0answers
50 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
votes
1answer
28 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
votes
1answer
110 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
votes
2answers
81 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
votes
1answer
169 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
306 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 ...
10
votes
2answers
278 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 ...
0
votes
2answers
119 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
votes
2answers
61 views

Equilateral triange, sum…

Just a short question: In a triangle we have $\sum \left(\frac{a}{b+c}\right)^{2}$. Is the triangle equilateral? I have derived
3
votes
2answers
277 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
votes
1answer
37 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 $180^{\circ}$
0
votes
1answer
50 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
votes
1answer
136 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
35 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
votes
0answers
81 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
votes
1answer
128 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
vote
1answer
58 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
votes
1answer
38 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
vote
1answer
59 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
vote
1answer
47 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
votes
1answer
73 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 $CH$, $AD$ ...
11
votes
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
vote
2answers
52 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
53 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
70 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 ...
2
votes
3answers
124 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
55 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
votes
1answer
22 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.
0
votes
1answer
43 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) $CD$
2
votes
2answers
126 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
votes
1answer
42 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 ...
0
votes
3answers
46 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
234 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
votes
1answer
49 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
votes
3answers
44 views

Trigonometric problem: Elevation angle [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
18 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
112 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 ...
0
votes
1answer
30 views

Finding angles in Barycentric system

How to find the angles of a triangle given the barycentric coordinates of its corners? Does it work if i take the first two components of every coordinate, and find the angles in the triangle (on the ...
2
votes
1answer
59 views

Geometry and Triangles

The triangle $ABC$ is such that $AB = 12cm$ and $AC = 8cm$. $X$ is the midpoint of the base $BC$. If the area of the triangle is $72 cm^2$ what is the length of the perpendicular from $X$ to $AB$ ...
0
votes
2answers
43 views

Find the value of EF and AC.

In the figure given below, BA, FE and CD are parallel lines. Given that AB = 15 cm, EG = 5 cm, GC = 10 cm and DC = 18 cm. Calculate EF and AC. I think the answer is EF= 8.66 and AC = 25.66 but I ...
1
vote
1answer
57 views

Prove that two triangles are congruent

${ABC}$ and $A'B'C'$ are two triangles. Let $P$ be the midpoint of $BC$ and $P'$ the midpoint of B'C'. Also, $|AP| = |A'P'|$ and $|AC| = |A'C'|$ and $\angle CAB$ = $\angle C'A'B'$. $2|AP| > |AC|$ ...