For questions about properties and applications of triangles

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3
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0answers
31 views

Number of triangles created after $n$ folds of a square

My daughter's grade 8 math homework included the following question. We were unable to find an answer, and I think we must have misinterpreted the question, as it seems way too hard. Fold a ...
0
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1answer
22 views

Angles inequality in acute triangle [duplicate]

Let $\alpha$, $\beta$, $\gamma$ be angles of acute triangle. How to prove that $(\tan(\frac{\alpha}{2}))^2 + (\tan(\frac{\beta}{2}))^2 + (\tan(\frac{\gamma}{2}))^2 \ge 1$? Does left side of equation ...
0
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3answers
73 views

Find unknown vertex of triangle given area and other 2 vertices

I need to find the coordinates of the 3rd vertex of a triangle given that I know the other 2 vertices and the area. The triangle is not guaranteed to be of any particular type (right, isosceles, ...
2
votes
2answers
103 views

Geometry: Find angle x in triangle

I have not been able to find a euclidean geometry solution to this, but any other solutions are also appreciated. Let ABC be a triangle with AB=CD and angles as marked in the diagram. Find the ...
0
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3answers
61 views

Point inside the area of two overlapped triangles

The question is as simple as that, but I have been trying to figure out an answer (and searching for it) with 0 results. I mean, given two triangles (in 2D) I want to find just a single point which ...
4
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7answers
103 views

explaining $|a+b|≤|a|+|b|$ in simple terms

I'm struggling to get to grips with the Triangle Inequalities. The problem is I don't really understand what it means. This is what my lecturer has written in the notes: $$ |a+b|≤|a|+|b|. $$ First of ...
4
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2answers
52 views

Bisecting the area and perimeter

In triangle $ABC$, $AB=16$, $AC=15$, and $BC=13$. Point $D$ is on $AB$, and point $E$ is on $AC$ so that $DE$ bisects both the area and perimeter of triangle $ABC$. (In other words, both $DA+AE$ and ...
0
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2answers
29 views

Right Triangle Theorem/terminology

Given a line segment $s$, there are exactly two right triangles which have $s$ as a hypotenuse. Is there a name for this theorem? Assuming this line segment lies on a Cartesian plane, how can we ...
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0answers
32 views

Concave triangle?

I know that in Euclidean and (I think) Spherical geometries don't have concave triangles, but is there any set of axioms that would allow this?
2
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4answers
126 views

|a-b|≤|a|+|b| is always true? [duplicate]

I wonder if |a-b|≤|a|+|b| is always true. I think it is true, but I don't see how to prove this mathematically. Thanks.
1
vote
1answer
34 views

Heron's formula to solve a triangle; any faster method?

A triangle has an area of $200cm^{2}$. Two sides of this triangle measure 26 and 40 cm respectively. Find the exact value of the third side. I used Heron's formula to solve this equation, but it ...
0
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1answer
27 views

Finding the coordinates of the third point in triangle

How would you find $x$ and $y$ coordinates of the third point in triangle($A$, $B$, $C$), if you know coordinates of $A$ and $B$, and angles at $A$ and $B$?
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2answers
35 views

Right angled median intersection question

All the information is included in the image. Find the length AB. Only clue I have is that the length CX is 2.5 where X is the perpendicular foot of B which i found out geometrically. However I ...
1
vote
2answers
32 views

Triangle perimeter proof

Given two points, $a, b$ inside a triangle $T$ whose perimeter is L we want to prove that the distance between them: $$|d(a,b)| \leq L. $$ One could easily prove this graphically, ie. the distance ...
0
votes
1answer
91 views

Find the area of the shaded region

This is the Figure, $ABCD$ is a square , $AB = BC = CD = DE = 21cm$. $AC$ and $BD$ are the diagonals of the square. The two semi circles are drawn with $AD$ and $BC$ as diameter. Find total are of ...
1
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1answer
47 views

Maximum length of projections

In triangle $ABC$, $BC=115$, $AC=127$, and $AB=89$. Let $P$ be a point varying on the cirucmcircle of triangle $ABC$. Let $M$ and $N$ be the feet of the perpendiculars from $P$ to $AB$ and $AC$, ...
0
votes
2answers
62 views

Why doesnt the gcse syllabus allow us to use herons formula?

I saw the answer to this question, it wants us to find the angle A using the cosine rule and then use the formula 1/2 ab Sin A to find the area. Why can't we just use herons formula - Area = (P ...
2
votes
1answer
29 views

Triangle/Geometry question

How do I solve this triangle question? In the figure below $\Delta OAB$ has an area of $72$ and $\Delta ODC$ has an area of $288$. Find $x$ and $y$.
1
vote
1answer
26 views

When using the Pythagorean theorem with a triangle, how do you know which numbers go where in the theorem?

When using the Pythagorean theorem with a triangle, how do you know which numbers, and x, go where in the theorem? For example, if I have a right trianle with the sides of $150$, $170$ and $x$, where ...
1
vote
1answer
46 views

Ravi substitution in inequalities

There is a well-known substitution for proving geometric inequalities: If $a,b,c$ are the side lengths of a triangle, then in an inequality involving $a,b,c$ it is possible to replace $a,b,c$ by ...
1
vote
1answer
36 views

Edges of what kind of graph may not be partitioned as triangles?

I would like to know edges of what kind of graph may not be partitioned as triangles? As an example edges of one of these graphs $K_7 , k_{12} , K_{3,3,3} , K_{5,5,5}$ may not be partitioned as ...
2
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1answer
29 views

Spherical Triangle

I know that the area for a spherical triangle is calculated as Area $= r^2(a+b+c-\pi)=r^2E$ where $E= (a+b+c-\pi)$ is the spherical excess I was wondering why do you have to multiply by $r^2$ (the ...
0
votes
1answer
38 views

jensen inequality in trigonometry [duplicate]

Can anyone help me how to prove $\sin A + \sin B + \sin C \leq \frac{3}{2} \cdot \sqrt[2]{3} $ I have idea use jensen but how to use it here?
3
votes
2answers
50 views

Finding angles plane geometry

$\Delta ABC$ is obtuse on $B$ with $\angle ABC = 90 + \frac{\angle BAC}2$ and we have a point $D \in AC$ (in the segment, I mean D is in between A and C) such that $\angle BDA = \angle ABD + ...
3
votes
4answers
63 views

Limit of ratio of areas of triangles defined by tangents to a circle

Let $AB $ be an arc of a circle. Tangents are drawn at $A $ and $B $ to meet at $C $. Let $M $ be the midpoint of arc $AB $. Tangent drawn at $M $ meet $AC $ and $BC $ at $D $, $E $ respectively. ...
0
votes
1answer
26 views

Point in a triangle plane determining any angles

Let $\triangle{ABC}$ be an arbitrary triangle. Is it true that for any angles $\alpha, \beta,\gamma\in [0,2\pi]$ with $\alpha+\beta+\gamma=2\pi$ one can find a point $M$ in the plane of the triangle ...
0
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0answers
20 views

Finding a specific weight triangle in a graph

It is possible to find the minimum weight of the triangles in a graph by using the following: Let G = (V, E; w) with w : V ∪ E → {−W, . . . , 0, . . . , W} ∪ {∞}, and V = {1, . . . , n}. Set D = ...
1
vote
2answers
27 views

Ratio of parts of a triangle

In the diagram above, segment DE is parallel to segment BC and the ratio of the area of triangle AED to the area of trapezoid EDBC is 1:2. How can I find the ratio of AE to AC? So far, I got the ...
3
votes
2answers
106 views

Proof of a geometric statement

If $D$ is a point inside a triangle $\triangle ABC$ then how the following statement is true. statement: $AB+AC>BD+DC$. I have tried in the following way but it seems to me defective. ...
0
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2answers
49 views

Trigonometry - how to find angles of triangle within another triangle?

What are the angles of angle 1 and 2? I don't see how any of them could be corresponding angles... The adjacent side of angle 2 is parallel to the hypotenuse of the bigger triangle, just to make ...
0
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0answers
41 views

Angle of Sine wave

How you do calculate angle of sine wave? Here in this example you can see the angle as the sine wave goes either side of the graph http://www.mathopenref.com/triggraphsine.html. For producing the sine ...
0
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1answer
422 views

Calculate 3rd point of a triangle, given 2 points and all angles in 2D

I have stumbled upon an interesting problem. I tried to find an answer here but there are just too many similar threads which did not really help me, so I was trying to figure it out by myself. The ...
0
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2answers
20 views

Name for line segment parallel to triangle base

In describing an elegant construction of a regular pentagon, i'm struggling to find a nice way of describing the following: A line segment starting at a point partway up one side of a (in this case, ...
0
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2answers
40 views

Determine if a triangle is right angled with only coordinates

What is the easiest way to determine that, given 3 coords, they DON'T form a right angled triangle? EG, (0, 0, 0), (0, 1, 0), (1, 0, 0) - forms a right angled triangle (0, 0, 0), (0, 1, 0), (1, 0.5, ...
1
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1answer
20 views

How to find a function that maps an element to row number in a triangle of integers?

This is a triangle of integers 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 ........ Is there some function that could map ...
0
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0answers
25 views

Quantifying the similarity of two line segments with a third line segment

In the program I'm developing, there are a large number of lines, and one point. One of the lines will split into two lines, the first line beginning with the original's first point and ending with ...
2
votes
2answers
34 views

The area of a triangle is $54\sqrt{6}$ square units. Find the lengths of the sides: $5x,6x,7x$

I realize I can use Heron's formula for this question. I did $54\sqrt{6}=\sqrt{9x\cdot 3x\cdot 4x\cdot 2x}$ but from there I must have done something wrong. Thanks for the help.
2
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0answers
31 views

Calculate the area of a triangular field, knowing that two and 1 angle.

Hello so this problem came up while I was studying trig. and I seem a bit stuck: Calculate the area of a triangular field, knowing that two of its sides measure $80$ m and $130$ m and between them is ...
-1
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1answer
33 views

Problem involving rhombus and its diagonals and height

If I know that one of the heights of a rhombus splits its longer diagonal in 2 segments equal to 7 and 11, how can I find the length of the base of the rhombus?
8
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4answers
346 views

Concentric Equilateral Triangles

I'm currently researching a particular dynamical system that is very geometric in nature. As part of this, I need to prove the following results (the second obviously implies the first). They are ...
2
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2answers
48 views

Simplest proof that the edge of an inscribed equilateral triangle bisects the radius

Context: I am giving a short talk on the Bertrand Paradox to a mixed group, many of whom have studied mathematics at a higher level some years ago. The point of the talk is the philosophical ...
0
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1answer
30 views

How to prove that perpendicular bisector of line joining altitudes also bisects the side?

Let ${D,E,F}$ be the feet of the altitude from ${A,B,C}$ in a ${\triangle{ABC}}$. Prove that the perpendicular bisector of ${EF}$ also bisects ${BC}$.
3
votes
1answer
31 views

Forbidden zones for circumcenter

Given a triangle $ABC$, let $A'$ be the middle point of $BC$, $B'$ the middle point of $AC$ and $C'$ the middle point of $AB$. It is well-known that the circumcenter of $ABC$ is the orthocenter of ...
4
votes
2answers
66 views

Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$

Let $a$, $b$ and $c$ be the three sides of a triangle. Show that $\frac{a}{b+c-a}+\frac{b}{c+a-b} + \frac{c}{a+b-c}\geqslant3$
0
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1answer
36 views

In a right triangle ABC with AH as height prove that BC, BA, BH form geometric progression.

Let's have a right triangle ABC(A=90 grades) and AH as height. Prove that BC, BA, BH form a geometric progression. Well is saw that they were similar triangles and showed that $$BC/AC=AC/HC=AB/AH$$ ...
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vote
2answers
59 views

Is there an equation that represents the nth row in Pascal's triangle?

I'm doing binomial expansion and I'm rather confused at how people can find a certain coefficient of certain rows. For example, if a problem was $(2x - 10y)^{54}$, and I were to figure out the ...
2
votes
2answers
59 views

How to find the position of point S on a tetrahedron if all segments are known?

I have been having this problem at work in this software I am writing. This question looks like a homework question but it is not... I promise. I took my problem and generalized it to a more simple ...
0
votes
2answers
60 views

Have to use pythagoras theorem

In parallelogram ABCD, the diagonals AC is at right angles to AB.If AB=12 & AC=13. I have to find the area of parallelogram. How can I use Pythagoras theorem here? I do not understand.
0
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1answer
59 views

Observations on integer-sided right triangles

Prove that in any integer side right angled triangle the following hold true: one side is always divisible by 3 one side is always divisible by 5 the product of two legs ...
0
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1answer
30 views

How to find the height of a 2D coordinate on a 3D triangle?

I would like to know how to dertermine the $Y$ coordinate of a point $M(X,Y,Z)$ in a triangle according to $MX$ and $MZ$, A,B,C ? Do I have to find the normal ? I have the coordinates of the points ...