For questions about properties and applications of triangles

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2
votes
2answers
331 views

Sine defined for a triangle inscribed in a circle with a diameter of one

Let a circle be drawn with a diameter of one (and thus a radius of one half). Then let a triangle with vertices A, B, and C be inscribed in the circle (i.e. points A, B, and C are arbitrary points on ...
3
votes
3answers
156 views

Prove Parallelogram Area Is Twice Triangle Area

I thought this would be easy but I can't seem to find the answer. Edit: I did my best to draw the diagram: $\overline{EC}=\frac{1}{3} \overline{AC}, \overline{AF}=\frac{1}{3} \overline{AB}, ...
1
vote
2answers
46 views

issues with geometry triangle

$4$ line drawn parallel to base of triangle such that they are equidistant.if the area of the most bottom part is 4 sq cm. find area of triangle? MY THOUGHTS : being weak in geometry i couldn't make ...
0
votes
6answers
330 views

New area of triangle if sides are halved

My question is that if we have a triangle, and we divide each of the side by 2 to get a new triangle, what will be the area of the new triangle in context to the original triangle? Please provide a ...
1
vote
1answer
188 views

How do I solve for the height of a triangle?

The basic triangle looks something like this: How do I solve for $h$? As an example, in one problem I was given $b = 45, c = 42, \angle C = 38^\circ$ I understand how $h$ divides $\triangle ABC$ ...
3
votes
2answers
73 views

Question about pythagorean triples

Given a,b,d natural numbers. Suppose (a, b) are two legs of a pythagorean triple. Also suppose (a, b+d) are two legs of another pythagorean triple. I'm looking for a way to show that given the ...
0
votes
2answers
296 views

Solving all possible triangles?

So we're doing oblique triangles -- Law of Sines and all that good stuff =). I have a bunch of problems that ask you to solve for "all possible triangles that satisfy the given conditions". For ...
3
votes
1answer
43 views

Odd and Even Triangles

I am about to make a report on the topic of characterization of line graphs then I came across the terms "odd triangles" and "even triangles". Does anyone know what these terms mean? To elaborate, I ...
3
votes
0answers
62 views

Howto prove that $\sum_{cyc}\cos\frac{A}{2}\cos\frac{B}{2}\le\frac{1+2\sqrt{2}}{2}+\frac{7-4\sqrt{2}}{R}r$

let $ABC$ is a triangle with inradius $r$ and circumradius $R$. Show that ...
0
votes
3answers
69 views

Can you find the the various values of a non 45-45- 90 triangle if only given hypotenuse and right angle?

I'm working on some homework. With a simple yes or no if you have a right triangle ABC with B being the 90 degree angle and not a 45-45-90 triangle and you have the value of the hypotenuse and the ...
0
votes
1answer
53 views

Determine missing angle in polygon

I'm trying to figure out this question: Determine the measure of angle a I'm guessing $a=96\unicode{0186}$ using the following work: $$a = 180 - 84 = 96 $$ ...
1
vote
0answers
43 views

Ratios of right triangle integer multiples to PI

It is known that in a right triangle with angles 30 and 60 degrees the cathetus at the 60 angle is equal to the 0.5 of hypotenuse. In other words an angle with cosine 0.5 is equal to PI/3. Is there ...
0
votes
3answers
142 views

Finding the Rate of distance between hands of clock

First, I think I don't understand the problem which asks about the greatest rate of change in distance between the tips of the hands of clocks. Does it mean where the increasing of distance is the ...
0
votes
1answer
173 views

Do degenerate triangles count? (2014 AMC 12B #12)

The problem is this: A set S consists of triangles whose sides have integer lengths less than 5, and no two elements of S are congruent or similar. What is the largest number of elements that S can ...
2
votes
1answer
109 views

Is it possible to reconstruct a triangle from the midpoints of its sides?

Take $ABC$ an arbitrary triangle, it is easy to take the midpoints $P$, $Q$, $R$ of sides $AB$, $BC$, $CA$, and we all know that the medians $CP$, $AQ$, $BR$ intersect at a point called the centroid ...
5
votes
3answers
172 views

An inequality for sides of a triangle

Let $ a, b, c $ be sides of a triangle and $ ab+bc+ca=1 $. Show $$(a+1)(b+1)(c+1)<4 $$ I tried Ravi substitution and got a close bound, but don't know how to make it all the way to $4 $. I am ...
2
votes
0answers
223 views

Triangle Packing-Problem

Theory and Question We define a normalized triangle $T$ as an ordered list of six points s.t. $p \in [0,1)$ for all $p \in T$. Let $T = [x_0, y_0, x_1, y_1, x_2, y_2]$ be a normalized triangle. We ...
1
vote
1answer
40 views

Given 3 points and there distances from eachother find a fourth point equidistant to the 3.

This question can also be asked: given a triangle, and its dimensions, whose vertices lie on the edge of a circle find the radius of the circle. I am not actually sure if there is enough information ...
2
votes
1answer
224 views

Conclusion from trigonometric identity

Let $\alpha$ and $\beta$ be angles in triangle, i.e $\alpha, \beta \in \left(0,\pi\right)$ can we conclude that $\alpha = \beta$ if the following statement is true: $$\left(\frac{\sin \alpha}{\sin ...
1
vote
0answers
13 views

3-D evaluations of a triangle

We all do evaluations of triangles on 2-D space based on the fact that the sum of its internal angles is 180 degree. When we draw a triangle on a sphere this sum changes and gets bigger than 180 ...
0
votes
2answers
96 views

Finding the length of the opposite and adjacent sides of a triangle

I am writing a small game in javascript. It's been a while since I have done any basic maths and I can't get some of my positioning to work properly. Apologies if this question is too simple, but I ...
0
votes
1answer
15 views

Position Value in a Triangle

I have a triangle. Each of the 3 corners is assigned a different value. Lets say corner 1 is 100, corner 2 is 200, and corner 3 is 300. I want to pick a coordinate in the middle of the triangle and ...
0
votes
1answer
188 views

What is the maximum number of triangles in a planar graph with n vertices?

The answer is obvious for small numbers of nodes: $$n<3: 0\\ n=3: 1\\ n=4: 3\\ n=5: 5 (see below)$$
3
votes
2answers
3k views

when to use sine vs cosine vs tangent

I'm a little confused about how you choose to use either sine or cosine or tangent over the others. Are they interchangeable given the same information you have about a right triangle? What are the ...
2
votes
1answer
129 views

Area of triangle inside triangle

In triangle $ABC$ we choose 3 points $D,E,F$, such that $\overline{AD} = \frac 13 \overline{AB}, \overline{BE} = \frac 13 \overline{BC}, \overline{CF} = \frac 13 \overline{CA}$. Draw segments ...
0
votes
2answers
75 views

Show that if dist(P, ℓ1) = dist(P, ℓ2) then P is on the bisector of the angle

In the image above line ℓ1 refers to $AQ$ and the ℓ2 refers to $AR$. It is given that dist(P, ℓ1) = dist(P, ℓ2). I think I have to show that $PA$ is the bisector of $\angle A$ but I don't know where ...
3
votes
2answers
2k views

How do you find the height of a triangle given $3$ angles and the base side? Image given.

This question has me absolutely stumped. This is the image of the question, how can I work out $x$? I've been doing a variety of attempts but I just cant get it.
0
votes
1answer
410 views

How to solve bearing of oblique triangle

I'm having a hard time finding the solution of the bearing given in our example. Our Example: Suppose there's a triangle with points named A,B, and C. Point A is named Bacoor. Point B is named San ...
0
votes
1answer
98 views

Calculate side of a triangle with known base and median

Is it possible to calculate the size a of the triangle where t is a median? If so, how?
0
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3answers
135 views

Guessing the nature of a triangle if one angle is 60 degrees

In triangle ABC, AB=12cm, angle B=60 degrees, the perpendicular from A to BC meets it at D. The bisector of angle ABC meets AD at E. Then E divides AD in the ratio $3:1,6:1,1:1,2:1$? I assumed it ...
-2
votes
1answer
40 views

Distance to the point of intersection of two altitudes in a triangle

Problem $P,Q,R$ are the points $(4,2)$, $(2,1)$ and $(6,-3)$. Also, $PS$ and $QT$ are altitudes in this triangle. A) Find the equations of PS and QT My answer: PS: $y-x-2=0$, QT: $5y-2x-1=0$ ...
0
votes
1answer
191 views

How to identify opposite and adjacent parts of right triangle

If you have a right triangle and both the opposite and adjacent sides have values of ex.10 or the same value. How do you determine which side is the opposite and which is the adjacent if they are both ...
1
vote
0answers
83 views

Area of a triangle using vectors

I have to find the area of a triangle whose vertices have coordinates O$(0,0,0)$, A$(1,-5,-7)$ and B$(10,10,5)$ I thought that perhaps I should use the dot product to find the angle between the ...
0
votes
1answer
137 views

Cartesian equation of a triangle

I am wondering what will be the equation of a triangle with vertices at (1,0), (0,-1), and (2,-1)? I really appreciate your quick responses on this! Shah
0
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2answers
18 views

angle in triangle of pre-known measure

I'm facing a problem. We have lengths of 3 segments. How to see if the triangle built of our 3 segments has a specific angle, for example 60°?
0
votes
1answer
239 views

Prove vertex-orthocenter distance is twice of side_midpoint-circumcenter distance

$AL$ , $BM$ , $CN$ are altitudes , $TD$ , $RF$ , $ES$ are perpendicular bisectors of sides. How to prove $AQ = 2PD$ ? By similar triangles $\triangle AQN \sim \triangle CQL$ and $ \triangle CQL \sim ...
1
vote
2answers
105 views

Triangle incenter relation

Let $ABC$ be a triangle in which $AB = AC$ and let $I$ be its in-centre. Suppose $BC = AB + AI$. Find $∠BAC$. I do not see how to start even, please help.
2
votes
2answers
134 views

When is $Ar(APD)=Ar(ABCD)$?

This question arose while I was answering this question, (we need to show $Ar(\Delta APD)=Ar(ABCD)$). First the original question: $ABCD$ is a quadrilateral. A line through $D$ parallel to $AC$ meets ...
2
votes
0answers
187 views

Computing Euler Angles from Direction Cosines Vector

My problem simply as the following: Suppose we measured the orientation of a plane object with respect to a reference fame. (where the reference frame parallel to plane frame). The unit normal vector ...
1
vote
2answers
113 views

Two parallelograms are equal in area.

I tried this question by constructing a line $PD$ therefore forming two triangles $ADP$ and QDP but couldn't establish the congruency relation between the triangles. My approach was that if I have ...
2
votes
7answers
462 views

Area of a triangle

The following problem in elementary geometry was proposed to me. As a mathematical analyst, I confess that I can't solve it. And I have no idea of what I could do. Here it is: pick a triangle, and ...
1
vote
2answers
379 views

Area of a quadrilateral inside right angled triangle

$ABC$ is right angle triangle. $AB=24 cm$, $BC=10 cm$, $AC=26 cm$. Point $D$ on $AC$ (hypotenuse) bisects $AC$ and connects point $E$ on side $AB$ such that $ED$ is perpendicular to $AC$. Side $AC$ is ...
0
votes
2answers
75 views

How $\pi$, $3.1415…$ and $180^o$ are adaptive together?!

I planed following to compute the circle's circumference. The circle's circumference finally can computable from: $$\lim_{\alpha\to0}{\frac{360^o}\alpha d} = 2\pi r$$ I don't want to follow above ...
2
votes
1answer
131 views

Distances to line passing through the centroid of triangle

Let $p$ be a line that pass through the centroid of a triangle $ABC$. Unless the line pass through one vertex, then $2$ verices are one side of the line, while the third one is on the other side. ...
0
votes
2answers
63 views

Question on inscribed equilateral triangle

Question: $ABC$ and $ODE$ are equilateral triangle with $BC || DE$. If $O$ is the center of the circle, then find the ratio $AQ:QC$ So, my thought on this is that, since we are not given the ...
2
votes
3answers
452 views

Find radius of a circle which is tangent to three known lines

I need to find the equation for a circle which is tangent to the following three lines: y=0 x=0 y=-x+0.338334 For the last tangent line equation, I know that it is tangent at the point (0.169167, ...
1
vote
4answers
194 views

Using the Law of Sines to find all triangles with given values of two sides and an angle

Our teacher skimmed over this and we have homework over it. Textbook is mostly unhelpful. I'm confused on how ambiguous case works, and everything I see online just confuses me more. I'm not quite ...
1
vote
1answer
575 views

equality of triangle inequality

$z$ and $w$ be nonzero complex numbers. How do I show that $|z+w|=|z|+|w|$ if and only if $z=sw$ for some real positive number $s$. I approached this by letting $z=a+ib$, and $w=c+id$, and kinda ...
3
votes
1answer
110 views

slice up a slice of a triangle into n areas of equal size

Figure description: The point $(0, 0)$ is in the upper left corner. The coordinate system grows to the lower right corner. The short sides of the big triangle have the same length. I want to slice ...
1
vote
2answers
64 views

Formula for sides of a triangle where the Perimeter equals to the Area [duplicate]

I was wondering if there is a formula that could generate the values of the sides of a triangle where his area equals to his perimeter. I only found that if the triangle is equilateral then ...