For questions about properties and applications of triangles

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2answers
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A box contains 5 rods whose lengths make triangles.

A box contains five rods whose lengths are 1", 3", 6", 10", 15". How many different obtuse triangles can be made using only three rods at a time. I determined that the answer is 1 because the ...
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0answers
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More generalization of the Sawayama lemma

Let $ABC$ be a triangle, $P$, $Q$ be two isogonal conjugate. $AP$, $AQ$ meets (ABC) at $D, E$ respectively. Two lines through $D, E$ meet (ABC) at $T, N$ and meet BC at $G, H$ respectively. Let $PG, ...
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0answers
14 views

Start and endpoint of line, creating arrow heads [on hold]

I have a start point(5.6,4) and an endpoint (6.1,3.15) I want to make an arrow head at the start point that is an equilateral triangle(60 degrees) with a length of .1. How can I accomplish this? ...
0
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1answer
16 views

Plotting triangles based on a single point with distance and angle.

I'm tasked with creating an arrowhead within a pdf program. I have a single point with at $x=5.6$, $y=4$ this would be point A of my triangle I want to make the sides equal at $90$ degrees angles ...
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0answers
50 views

Right angled triangle and Pythagorean triplet

Show that there exists a right angled triangle with rational sides and area $d$ if and only if $x^2,y^2$ and $z^2$ are squares of rational numbers and are in arithmetic progression with common ...
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3answers
53 views

How to find the area of the following isosceles triangle

I am stuck with the following problem : What is the area of an isosceles triangle whose equal sides are $20$ cm and the angle between them is $30^{\circ}$ ? It is a nineth standard problem and ...
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0answers
25 views

A generalization of the Sawayama lemma

Let $ABC$ be a triangle, let $D$ be a point on the line $BC$. The Thebault circle is a circle tangent $AD, BC$ and the circumcircle (yeallow circles in the following figure). I give a ...
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1answer
18 views

Sides of a triangle are in Arithmetic Progression, then find $\tan (\alpha+ \frac{\beta}{2})$

The sides of a triangle are in Arithmetic Progression. If the smallest angle of the triangle is $\alpha$ and largest angle of the triangle exceeds the smallest angle by $\beta$, then find the value of ...
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3answers
28 views

Geometry problem related to right angled triangles.

In the given figure $AC = 12 cm, AE = 6 cm$ and $CD = 8 cm$. CD is perpendicular to $AD$ and $BE$ is perpendicular to $AC$. How can we find the value of $BE$?
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2answers
44 views

How are the trigonometric ratios geometrically defined for non-acute angles?

The usual way trigonometric ratios are geometrically defined is always relative to an acute angle. So this way inside a right triangle, the trigonometric ratios are defined by the ratios of hypotenuse,...
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2answers
49 views

Equilateral triangles

Let $ABC$ be a triangle with $AB = 1$, $AC = 2$ and $m(\widehat{BAC}) = 30^\circ$. We build on the outside the equilateral triangles $ABM$ and $ACN$. Let $D$, $E$ and $F$ be the midpoints of $AM$, $...
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1answer
45 views

Problem on Equilateral Triangle and points

Equilateral $\triangle{ABC}$ with sides $2\sqrt{3}$. Let $P$ be the point outside$\triangle{ABC}$ such that points $A$ and $P$ lie opposite to $BC$. Let $PD$, $PE$, $PF$ be the perpendicular dropped ...
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1answer
22 views

Tracing the sides of an equilateral triangle

Is there any way I can get the points in 2D plane on the sides of an equilateral triangle for certain infinite animation sequence? For example in case of tracing the circumference of the circle, I ...
0
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1answer
34 views

Sides of triangle are in A.P., find its perimeter

The sides of a triangle are in Arithmetic Progression $(A.P.).$ If the smallest angle of the triangle is $\alpha$ and largest angle of the triangle exceeds smallest angle by $\beta$ , then what is the ...
1
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1answer
43 views

How do I find the partial derivatives of heron's formula?

Heron's formula finds the area $A$ of a triangle with sides of length $a$, $b$, and $c$: $$A=\sqrt{s(s-a)(s-b)(s-c)}$$ where $s$ is the semiperimeter of the triangle: $$s=\frac{a+b+c}{2}$$ How do ...
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0answers
39 views

Trigonometric roots of a cubic

Let the product of the sines of the angles of the triangle is $\frac{2}{3}$ and the product of their cosines is $\frac{1}{9}.$ If $\tan A$ , $\tan B$ and $\tan C$ are the roots of the cubic, find the ...
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0answers
26 views

How to rewrite equation to get a quadratic patch

I would like to understand the given rewrite or transform from one equation to another. This is the original equation: $$p^*(q)=(u,v,w)\left( \matrix{q-n_i\big((q-x_i) \cdot n_i \big) \cr q-n_j \big((...
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1answer
35 views

“Easy” triangle problem (hight school)

Can someone give me a hint to this "easy" problem? In the triangle ABC, we have: DE || BC, FE||DC, AF=1, FD=2, find DB=?
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0answers
10 views

How do I determine the angles to cut 3 wooden 2x2's where they meet at top of a Tetrahedron? (Triangular pyramid)

I'm trying to build a Star Tetrahedron (merkaba) out of 4 foot long 2x2's. I already cut the 30 degree angles for the base of the first tetrahedron which formed a nice equilateral triangle but now I'm ...
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0answers
13 views

efficiency of different whole-number-mass-to-a-power in balancing a regular triangle/tetrahedron

I saw this qustion: http://puzzling.stackexchange.com/questions/186/whats-the-fewest-weights-you-need-to-balance-any-weight-from-1-to-40-pounds Suppose you want to create a set of weights so ...
0
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1answer
30 views

Triangle Inequalities in Right Angled triangle.

In $\triangle{ABC}$, $\angle{ABC}=90^{\circ}$, $AB=BC$ and $AC=\sqrt{3}-1$. Suppose there exist a point $P_0$ in the plane of $\triangle{ABC}$ such that $AP_0+BP_0+CP_0 \leq AP+BP+CP$ for all points $...
1
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1answer
32 views

The coincidence orthocenters of the two triangles

Let $CH -$ height in acute-angled triangle $ABC$. Some points $K$ and $N$ are on side $AB$. Let $O_1 -$ orthocenter of triangle $ACN$ and $O_2 -$ orthocenter of triangle $BCK$. Prove $$O_1=O_2=O \...
2
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1answer
60 views

Geometric arithmetic: triangular number triples [closed]

Call a triple $x, y,$ and $z$ of numbers triangular if and only if there is a triangle whose sides are in the triple ratio $x:y:z$. Since the sum of two sides of a triangle exceeds the remaining side, ...
2
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1answer
34 views

Identifying a triangle in the 3d-space as acute, obtuse, right or equilateral

Triangle $ABC$ has vertices $A(-1, 1, 3)$, $B(-1, 3, 5)$, and $C(-3, 3, 3)$. What kind of triangle is $ABC$? Justify your answer. So far all I have done is I found the distance between $AB$, $BC$ ...
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4answers
301 views

Show that a point is a midpoint of a side of a triangle

In $\Delta ABC$, the bisector of $\angle A$ intersects $BC$ at $D$. The perpendicular to $AD$ from $B$ intersects $AD$ at $E$. The line through $E$ parallel to $AC$ intersects $BC$ at $G$, and $AB$ at ...
4
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2answers
65 views

Solve linear system with $A_{i,j} = \langle e_i, e_j\rangle^2$, edges of a triangle

I have three vectors in $e_i\in\mathbb{R}^3$ that form a triangle. Let us consider now the linear equation system $Ax=b$ with $$ A_{i,j} = \langle e_i, e_j\rangle^2,\\ b_i = \langle e_i, e_i\rangle. $$...
3
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1answer
37 views

On constructing a triangle given the circumradius, inradius, and altitude .

I was recently pondering about constructing triangles given different attributes of it. I am wondering whether we could construct a triangle given its Circumradius $R$ , Inradius $r$, and length ...
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2answers
115 views

The problem of congruent areas in a triangle.

A problem was posed in front of me and I couldn't solve it after multiple attempts-- Consider any triangle and 3 concurent cevians are drawn from each of its 3 points . Now the figure formed has 6 ...
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2answers
53 views

Are there some undiscovered/unproved theorems about Euclidean triangles?

For a particular case of a figure called simplex, a triangle is surprisingly complicated (in my opinion). As an illustration, see the list of triangle topics on Wikipedia, and the Triangle page. The ...
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0answers
47 views

Finding angles in a triangle

If $AD$ is the median to side $BC$ of $\Delta ABC$ & $\angle B =2 \angle C$, then find $\angle B$ I feel something is missing in the question
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0answers
24 views

Finding angle in triangle [duplicate]

An equilateral triangle $\Delta ABC$ & $P$ is any point inside the triangle such that ${PA}^{2}={PB}^{2}+{PC}^{2}$, then $\angle BPC$ is - I am unable of how to think to do this question
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2answers
59 views

Finding angle in an equilateral triangular pyramid

Given an equilateral triangular pyramid (refer the below diagram) $\Delta ABC$ & $P$ is any point inside the triangle such that ${PA}^{2}={PB}^{2}+{PC}^{2}$, then $\angle BPC$ is - I am unable ...
3
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2answers
56 views

Triangle with $3$ unknowns

I have a situation where I am trying to calculate a leading shot for a character in a 2D top down game. The enemy character moves with a certain speed $s$, which is applied to its normalized ...
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0answers
25 views

pack equilateral triangle

I'm working on a problem of inscribing equilateral triangle for some time now and it goes like this : using only a foot rule and a compasses , show a way of inscribing an equilateral triangle into ...
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0answers
19 views

Angle for pitched roof on isometric drawing.

I have some experience 2D drafting/cad work and tried to do a simple house drawing in isometric view. I had trouble working out the roof and as you can see in the picture it looks rather flat. The ...
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3answers
36 views

Deriving of formula for finding the length of median

In the below image $AD$ is the median of $\triangle ABC$ We know that $m_A = \frac 1 2 \sqrt{2b^2 + 2c^2 - a^2} $ But can someone tell me how it's derived !! I am just unable to think of it !! ...
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2answers
33 views

Prove on Incenter and mid point.

Let the incircle (with center $I$) of $\triangle{ABC}$ touch the side $BC$ at $X$, and let $A'$ be the midpoint of this side. Then prove that line $A'I$ (extended) bisects $AX$.
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1answer
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Prove between Simson line & Nine point circle.

Prove that the Simson lines of diametrically opposite points on the circumcircle are perpendicular to each other and meet on the nine-point circle. I proved the first part of the problem but not able ...
2
votes
2answers
85 views

What is the range of $λ$?

Suppose $a, b, c$ are the sides of a triangle and no two of them are equal. Let $λ ∈ IR$. If the roots of the equation $x^ 2 + 2(a + b + c)x + 3λ(ab + bc + ca) = 0$ are real, then what is the range of ...
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0answers
49 views

Condition for the existence of a triangle

Could you please explain and solve this problem for me? I would really appreciate it. The more depth of explanation, the better. Let $a$, $b$, $c$ be non-collinear vectors. Show that the necessary ...
2
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1answer
28 views

Similar Triangles--Find the measurement of the unknown side [closed]

This is a question I know I got wrong on a final exam in a very easy class for teaching elementary geometry/prep for Praxis II. I actually received a 99% average in the entire course because of the ...
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1answer
67 views

A geometry problem hinting similarity of triangles .

I recently came across a geometry problem , published in an local magazine(publishing at high school and under graduate level) and was under Difficulty : Hard sub heading. Consider a $\triangle ABC$ ...
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3answers
36 views

Prove for Pedal & Isosceles triangle.

The tangents at two points $B$ and $C$ on a circle meet at $A$. Let $A_1B_1C_1$ be the pedal triangle of the isosceles triangle $ABC$ for an arbitrary point $P$ on the circle, as shown below. Then ...
2
votes
1answer
119 views

Unique Trianlge Count sequence

Consider a simple graph $G(V,E)$, such that $V = \{1,2,\dots, n\}$. We can define the triangle count of a vertex as follows: $\Delta(v) = $ Number of triangles in the graph such that $v$ is one of ...
0
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1answer
27 views

Counting number of points making angle < 90

I have a around 1000 points and 1000 segments in the form of $(x_1, y_1, x_2, y_2)$ meaning the segment starts at coordinate $(x_1, y_1)$ and finishes at $(x_2, y_2)$. For each line i want to know how ...
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1answer
43 views

Can you solve this geometric question on triangles? [closed]

In a triangle $ABC$, $D$ is a point on the side $BC$.Given: $AD=10$,$BD=DC=8$ and $BC*AD=6$.What is the length of $BC$? a.$5$ b.$10$ c.$15$ d.$20$ That was asked in a newspaper quiz.
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2answers
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Get second vertex of isosceles triangle [closed]

Given the equal sides of the triangle and the angle $\theta$ between them as well as the other 2 vertices of the triangle how do I get the second base vertex coordinates. Sorry for my poor drawing. <...
0
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4answers
59 views

Triangle - Trapezoid [Geometry]

I'm having trouble with following assignment: "Sides of triangle are $13$, $14$, and $15$. Line parallel to the longest side cuts through the triangle and forms a trapezoid which has perimeter of $39$...
3
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3answers
67 views

Prove: In a Triangle, $II_1 = a\cdot \sec \frac{A}{2}$

Prove that $II_1 = a\cdot \sec \dfrac{A}{2}$. $I$ is center of incircle, $I_1$ is center of excircle. What I did is : Drop $ID \perp AB$, & $I_1F \perp AF$ at $F$ So $ID\parallel I_1F$ $\dfrac{...
2
votes
1answer
70 views

Proving Gerretsen's Inequality

Today in class we were shown Gerretsen's inequality: $$16Rr-5r^2\leq s^2 \leq 4R^2+4Rr+3r^2$$ Where $R$, $r$, and $s$ are the circumradius, in radius, and semiperimeter of a triangle. After some ...