Tagged Questions

For questions about properties and applications of triangles

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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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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 ...
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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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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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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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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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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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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 ...
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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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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 ...
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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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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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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 ...
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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 ...
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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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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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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. <...
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 $39$...