For questions about properties and applications of triangles

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0
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1answer
36 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
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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
135 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
30 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
71 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
125 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
51 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
37 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
58 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
42 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
67 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
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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
vote
2answers
46 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
51 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
54 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
112 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
21 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
40 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
110 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
39 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
44 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
209 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
43 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
17 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
111 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
27 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
58 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
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2answers
37 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 $\angle CAB$ = $\angle C'A'B'$. $2|AP| > |AC|$ ...
0
votes
1answer
100 views

Solve an Angle-Side-Angle special case triangle if it has an obtuse angle?

I've seen this type of problem multiple times on homework, and it's confusing me like mad. The scenario: We have a triangle. It is a special case triangle, with one angle, one side, and another ...
3
votes
2answers
2k views

How to find opposite and adjacent lengths of a right triangle given the hypotenuse and angle?

I'm writing a few functions for a JavaScript game engine. Is it possible to calculate the length of the legs of a right triangle given ONLY the length of the hypotenuse and an angle?
1
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1answer
79 views

Is the centroid of any triangle can be calculated by averaging vertices

Is it always true that the centroid of any triangle can be calculated by averaging the x and y coordinates of its vertices without bothering with finding medians?
4
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3answers
95 views

Prove that this triangle is equilateral?

Given $\triangle ABC$. Let $D$ be the point where the altitude form the $A$ vertex intersect $\overline{BC}$ and the point $E$ is the intersect between the bisector of $\angle ABC$ with ...
2
votes
3answers
93 views

Finding circumcentre

Tangents are draw from $P(2,3)$ to $x^2+y^2=4$ meeting at $Q,R$ on circle. Parallelogram $PQSR$ is completed. Find the circumcentre of triangle $QSR$. My attempt: Clearly, the parallelogram is a ...
1
vote
1answer
45 views

Find out the $\angle PRQ$

please, help me to solve this.How can I proceed.I just need help. $PQR$ is a triangle. $M$ is a point on $QR$.here,$QM=1/3RM$ , $\angle RPM=30^ \circ$ and $ \angle QPM=20^ \circ$ now,$ \angle PRQ=??$ ...
1
vote
1answer
166 views

Orthocentre of a triangle defined by three lines

Problem: If the orthocentre of the triangle formed by the lines $2x+3y-1=0$,$x+2y-1=0$,$ax+by-1=0$ is at the origin, then $(a,b)$ is given by? I would solve this by finding poins of intersection and ...
0
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4answers
62 views

Mathematics based on triangles

How to find the third cordinate of a triangle , where as other two points are known. and a angle is known. Lets say , the two points are (0,0) , (600,0) and we need to find the third cordinate . ...
0
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1answer
19 views

Obtaining consistent triangle surface normals.

I am given 3 points in a random order like so... calculateSurfaceNormal(point1, point2, point3); I have implemented the method by simply saying... ...
0
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1answer
36 views

Change in length of a right triangle

And so my question is how do I prove the ??
3
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1answer
120 views

Locus of the centres of equilateral triangles (contest problem)

Given a triangle $A_0A_1A_2$ determine the locus of the centres of the equilateral triangles $X_0X_1X_2$ satisfying the condition that each of the lines $X_kX_{k+1}$, $k=0,1,2$ passes through ...
0
votes
1answer
401 views

Geometry/Programming- Draw An Equilateral Triangle Given One Point And A Desired Rotation

I feel this question has a stronger mathematical basis than strictly computer science. I am currently drawing an equilateral triangle given its center and its radius like so. I would like to ...
0
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2answers
50 views

Does the median make angles in the same proportion as the sides?

Till I remember I had studied this in the lower classes, but am not sure whether this is true or not. In the figure CD is a median. Does CD divide the angles 1 and 2 in the same ratio of the sides a ...
1
vote
0answers
31 views

How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
4
votes
3answers
100 views

Making 7 vertices triangle free graph bipartite by deleting an edge

Can anyone assert or refute the following claim? Claim: In every triangle free undirected graph $G=(V,E)$, $|V|=7$, there exist an edge $e\in E$ such that $G'=(V,E\setminus\{e\})$ is ...
0
votes
3answers
123 views

Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
1
vote
2answers
20 views

Solving a triangle using the given equation

In a triangle $ABC$ $2a^2+4b^2+c^2=4ab+2ac$ then the numerical value of $cos B$ equals ? ($a,b,c$ are sides opposite to angles $A,B,C$) I tried to use cosine rule , but couldn't adjust terms ...
4
votes
2answers
159 views

Unusual result when comparing trigonometry and Pythagoras in triangles.

I'm a Scottish Higher maths student. I was looking over some old textbooks, and came across a seemingly easy question, involving a circle within a triangle. I used the expected method to solve it; ...
1
vote
1answer
29 views

Computing distance in circle

It seems to me as pretty simple, but I just can't get hold of it: I am trying to compute fn(x, r). Thanks.