For questions about properties and applications of triangles

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1answer
80 views

Proving triangle inequality using complete-linkage between clusters and arbitrary dissimilarity measure

Assuming a dissimilarity measure d satisfies the usual properties, I need to prove that complete linkage ( i.e. d(A,B)=maxx∈A,y∈B{d(x,y)} ) either satisfies or does not satisfy the triangle inequality ...
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1answer
381 views

Geometry - optimal 2D mesh between X expendable points

Say you have X points on a plane. If you connect two points, you form a line. Connecting three points forms a triangle. A line cannot cross a line, and a smaller triangle cannot be created inside a ...
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1answer
18 views

Ratio of bisected cevian in triangle given intersection point

I have the coordinates of points $A$ $B$ and $C$ that form triangle $\triangle ABC$, and the coordinates of a point $D$ inside of $\triangle ABC$. Imagine a cevian, connecting points $A$ and $D$, and ...
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0answers
17 views

Find the lengths of sides of a right triangle

If I know the length of the hypotenuse of a right triangle ONLY, is it possible to find the lengths of the remaining 2 sides of that same triangle?
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33 views

Iterating three tangent circles using Malfatti Circles

First, construct three tangent circles (blue circles), then construct the triangle joining their centers. Then construct three Malfatti Circles for this triangle (green circles). Go on. What I'm ...
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2answers
28 views

Finding length and width of old square

The area of new shape is $A=130$ m$^2$. The original square had $2$ m added to its width and $5$ m to its length, the problems asks for one of the original sides (since they're all equal of course). ...
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1answer
28 views

Is it possible for two triangles to be different if the sides of one is equal to another?

I was reading Euclid's Elements E-book I found online and got stuck on this concept. I will just copy what I found to be very absurd. There could still be another different triangle with the same ...
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3answers
221 views

Finding area of sector inside an triangle

I have been asked this question from a junior and could not solve the question in a simple way. I am asking help on this platform. For a triangle $ABC$, Points $D, E$ on $AB$, where $AD:DE:EB=2:2:1$....
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0answers
30 views

What will be area of an equilateral triangle? [on hold]

my question is A side of an equilateral triangle is 24 root 3. Inside this triangle two other equilateral triangles is made such that there inner areas becomes same. find out side of smallest ...
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2answers
57 views

Solutions of triangles - proof

Question: For a triangle ABC, prove that: $$r_1 + r_2 + r_3 = r + 4R$$ Where $r_1,r_2,r_3$ represent the radius of the ex-circles opposite to angle A, B, and C respectively. $r$ represents the ...
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1answer
40 views

Let D be the midpoint of BC in triangle ABC. Let E be the midpoint AD, F be the intersection of line BE with side AC. Find $\frac{AF}{FC}$.

Let D be the midpoint of side BC in triangle ABC. Let E be the midpoint of line AD and let F be the intersection of line BE with side AC. Find $\frac{AF}{FC}$.
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1answer
379 views

A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
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2answers
36 views

Minimal perimeter of a triangle

Imagine a triangle with a base $[0, s]$ and a height $h$. ($s, h \gt 0$) For what orthocentre $x$ does the triangle have a minimal perimeter and how long is it? Now, the proof starts with: ...
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0answers
50 views

Find the length of the side of a right angle triangle inside a circle

Hello Stack Exchange. I have a question which has really been preventing me from making a certain program.In my program I need to find the length of AC using only AB and BD.The triangle is right-...
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2answers
25 views

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
14 views

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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1answer
2k views

Triangle dissection, no shared edges

It's possible to divide a triangle into smaller triangles such that no edge lengths are shared. Alternately, no two internal triangles share two vertices. The top three are the known simplest ...
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0answers
14 views

Start and endpoint of line, creating arrow heads [closed]

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? ...
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0answers
52 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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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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3answers
60 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
29 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
29 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
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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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1answer
455 views

Finding Y coordinate of third triangle point when X coordinate and two other points are already known

Suppose you know the coordinates for points A and B of a triangle. We can refer to those coordinates as (Ay,Ax) and (By,Bx). Also, suppose you know the X coordinate for point C (Cx) but do not know ...
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2answers
50 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
551 views

How to find heading angle to an object whose x,y coordinates are known?

Scenario: I have a map with a marked location on it. I know my x,y coordinates on the map (top left corner is 0,0), my distance from that marked location, my heading angle relative to true north (0 is ...
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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
23 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 ...
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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 ...
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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
28 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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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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1answer
468 views

For which $N$ is it possible to alter one side of an equilateral triangle of side length $N$ to get another triangle of integer side lengths, …?

This question is "inspired" by the Rupsa and Equilateral Triangle problem from Code Chef's "October Challenge 2015". The deadline of 12 October 2015 has passed. Given an equilateral triangle having ...
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1answer
36 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
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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 ...
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2answers
258 views

How Many Triangles are Created by n Lines in the Plane?

Suppose we are given n lines in the plane in "general position", which in the present case we define to mean the following: A. no 2 lines are parallel or identical B. no 3 lines have common ...
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1answer
33 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 $...
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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 \...
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1answer
61 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, ...
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3answers
4k views

How to find the type of triangle when given the ratio of it's sides?

Q.The sides of a triangle are in ratio 4 : 6 : 7, then the triangle is: (A) acute angled (B) obtuse angled (C) right angled (D) impossible It's definitely not (C) right-angled ...
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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 $$ A_{i,j} = \langle e_i, e_j\rangle^2,\\ b_i = \langle e_i, e_i\rangle. $$...
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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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1answer
40 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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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 ...
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4answers
158 views

In $\triangle ABC$, if $\cos A\cos B\cos C=\frac{1}{3}$, then $\tan A\tan B+\tan B \tan C+\tan C\tan A =\text{???}$

In $\triangle ABC$, if $$\cos A \cos B \cos C=\frac{1}{3}$$ then can we find value of $$\tan A\tan B+\tan B \tan C+\tan C\tan A\ ?$$ Please give some hint. I am not sure if $\tan A \tan B+\...
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2answers
117 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 ...