For questions about properties and applications of triangles

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Triangles with vertices on conics and their foci

Let $A$, $B$, and $C$ be the lengths of the three sides of a triangle. Let $α$, $β$, and $γ$ be the measures of the angles opposite those three sides respectively. Mollweide's formula tells us that ...
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Issue with a right-angled triangle

The area of the right angle triangle is $18\text{ cm}^2$ and the ratio of its legs is $2:3$. What is the length of the hypotenuse? I assumed the lengths of two sides to be $2x$ and $3x$. I used ...
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196 views

How to easily prove Euler's theorem, $OI^2=R(R-2r)$?

If $R$ is the circumradius and $r$ is the inradius of some triangle $ABC$, with its circumcenter being $O$ and incenter being $I$, then how to prove: $$OI^2=R(R-2r)$$ I have seen many mentions of ...
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Closest Points on Two Triangles in 3D Space

I have two triangles in 3D space, defined by 3 (x, y, z) points each. I'm looking to find the closest points between the two triangles, whether that be on surface, edge, or point. I'm unsure how to ...
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How to find the length of the union of Isosceles triangles

I am given N number of right angles triangles all of which are also Isosceles triangles. For each triangle, I am told where they start on a number line and where they end on a number line with end ...
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How to calculate normal (of magnitude 1) of a triangle?

I am currently doing a bit of geometry practice and wanted to know how to calculate the normal (of magnitude 1) of a triangle defined by 3 vertices: a, b and c`. ...
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How to prove that $FC/FA + GC/GA= 0$ from this triangle problem?

In triangle $ABC$, a transversal line intersects $AB$, $BC$, $CA$ at $D,E,F$ respectively. $BS$ intersects $AC$ at $G$, where $S$ is the intersection of $AE$ and $CD$. How to prove that ...
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123 views

How to prove these equations base on this following interior and exterior angle bisectors problem?

In the triangle $\triangle ABC$, length of $BC$ is larger than length of $AC$. The interior angle bisector of $\angle C$ intersects $AB$ at $D$; and the exterior angle bisector of $\angle C$ ...
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48 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 ...
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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 ...
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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 ...
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Largest possible value of a side

ABC is a triangle with side a, b,c with $a\geq b\geq c$ and $sin^3A+sin^3 B+ sin^3 C=a^3+b^3 +c^3$ How do I find the largest possible value of a? I tried to use the law of sines ratio, but it ...
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257 views

Finding Areas in triangles using ratios

What theorem/theorems should be used to find the shaded area? Y and M lie on the sides Ab and Bc respectively of the triangle YMB such that AY/MI= 1/4 and O/M = 1/3. Area ABC=35 PC and QA intersect ...
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88 views

maximum length of a scaled vector in a triangle (simplex)

Given a triangle (or, in general, a simplex) $T$ and a vector $\vec{s}$, I'd like to compute the quantity $$ \max\{|x-y|: x,y\in T, x-y = \alpha \vec{s}, \alpha\in\mathbb{R}\} $$ i.e., the maximum ...
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Two coloured plane

Can you prove that For any two angles $θ,ϕ$ there exists a monochromatic triangle that has angles $θ,ϕ,180−(θ+ϕ)$ in two coloured plane?
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Triangular exponentation logarithm and inverse

The generalized formula of triangular exponentation on real numbers field is $x ^ {\triangle y} = \frac {1} {y \cdot B (x, y)} = \frac {\Gamma(x + y)} {\Gamma(x) \cdot \Gamma(y + 1)} $ It's my ...
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571 views

General formula for computing triangular gaussian quadrature.

While this is a simple question, I'm totally lost. Is there any general formula for generation of n-point gaussian quadrature over a triangle? I'll use this formula to generate a variable-point (7, ...
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Pythagorean theorem question

In an isosceles triangle, the length of each leg is $13$ and the length of the base is $24$. What is the length of the altitude drawn to the base?
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Let $W1 = 0.5$, $W2 = 0.75$, and $\theta = 1$, find two vectors that satisfy $w\cdot x = \theta$.

Let $W1 = 0.5$, $W2 = 0.75$, and $\theta = 1$, find two vectors that satisfy $w\cdot x = \theta$. Can someone please guide me? I know I'm supposed to use $a \cdot b = \|a\| \|b\| \cos( \theta )$.
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42 views

How to find a triangle's perimeter only using base and height?

Without measuring the length of the other two sides, is there a way to find the perimeter with one side (Base) and the height of that side?
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Prove that angles are equal using complex numbers

that in triangle $\Delta ABC$, where $D$ is a point on side $BC$, $E$ is a point on side $AB$, $BD=AC$, $AD=AE$ and $AB^2=AC\cdot BC$: angle $BAD$ is equal to angle $CEA$. This problem can be quite ...
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Producing a 3D Net from a 3d inspired image

Producing a 3D Net from a 2D Image I'm trying to find the volume of the illustration, I've taken reference from the medium size of a strawberry's diameter, I've applied this scale to the remaining ...
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Proof for Pappus's Centroid Theorem with basic geometry?

How to prove Pappus's Centroid theorem about volume for a triangle rotated around an external axe? The theorem says that the volume V of a solid of revolution generated by rotating a plane figure F ...
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Using similar triangles, find l?

I am attempting to render some 3D shapes and am having trouble with some of the math. I'm hoping someone can help me out here. Given two arbitrary points in space, P and C, I need to find l. I have L ...
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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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Find the Radical Axis of the Circumcircle of Triangle ABC and its Nine Point Circle

Given a triangle ABC, find the radical axis of its circumcircle and its nine point circle.
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When proving two congruence how do we know which angle/letter comes?

Like lets say this is the picture Like how would i know which angle comes first for example After proving which angle congruent to which, how do i write. Triangle VTU =/congruent to CTB or Triangle ...
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22 views

How to find a valume of a prism when when we are not given the height?

So I was learning how to find the surface area and volume. I came across few youtube tutorials which were simple. And in my book I found much harder problem. It asks me to find the volume of the prism ...
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27 views

Prove that $\frac{t}{3} < Rr$.

The task is: An optional triangles area should be $t$ and its circumscribed circle radius $=~R$ and its inscribed circles radius $=~r$. Prove that $\frac{t}{3} < Rr$. I was trying to solve it by ...
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Cutting triangles into two pieces

The task is this: We cut an obtuse triangle into two pieces by cutting it from its largest angle vertex into two isosceles triangles. How large are its angles (of the original triangle) if we could ...
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Solving the Height of a Triangle at a Point

I just solved a right triangle with all the angles and measurements: I want to find the height of the triangle at 2ft: How would I find this and what would the height be?
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Triangle Inequality for vector spaces

Consider the triangle ABC and suppose that the lengths ∥AB∥ and ∥AC∥ of the directed ⃗⃗ line segments AB and AC are 2 and 3, respectively. What one concludes about the length ⃗⃗ ∥BC∥ of BC? ...
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Triangle bisector length problem

I had this task on my math test: In the triangle ABC, the bisector of angle at C intersects side AB at point D. Given that the angle at C is 120°, |AB| = 6cm and |BC| = 12cm, calculate the length of ...
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Proof of the Crossbar theorem

A teacher asked me to prove the well known Crossbar theorem. I tried it in the following way:- Given: If $D$ is in the interior of $\triangle ABC$, then prove that $\overrightarrow{AD}$ intersects ...
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Quadrilateral inside a Parallelogram

$ABCD$ is a parallelogram with diagonal $AC$. $BE$ is the median to side $CD$ ,intersecting $AC$ at $O$.If the area of $ABCD$ is $120$ units, find the number of square units in the areas of ...
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48 views

Is a 2-sided polygon in 2D a reasonable concept, and what is then the definition?

A 5-sided polygon can be reduced to a 4-sided polygon by removing one vertex and the two connecting edges, and re-connecting the two adjacent vertices with an edge, whereby the interior angle goes ...
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Create a topographic plot based on distance and angle measurments

I have a distance sensor that rotates over a +10:-10 degree arc, measuring distance (in cm) at each angle. This sensor is suspended above the ground looking downward. So my final data looks like this: ...
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How do we use Vector Method to find ratios α and β?

We have △$OAC$ with a point $B$ on $OC$, a point $D$ on $OA$, and a point $E$ on $CA$, such that: $OBED$ is a parallelogram. $OB=3BC$ $DA=αOA$ $EA=βCA$ Find $α$ and $β$ using vector method. I ...
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Maximize the distance to vertices in an equilateral triangle

Given a set which represents an equilateral triangle in a plane, find the point inside the set which maximizes the product of the distances to the vertices. Optimizing the product of the distances ...
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Show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$

If the internal bisectors of the angles of the triangle ABC make angles $\alpha,\beta,\gamma$ with sides $a,b,c$ respectively then show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$ I tried to ...
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Double integral over a triangle

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a smooth function (derivable, integrable over all of $\mathbb{R}^2$). Let $T$ be a triangle in $\mathbb{R}^2$, defined by its vertices : $A=(x_a,y_a)$, ...
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Definition of triangle

If a polygon has 3 sides, but one side has zero length (or one angle is zero degree), is it still a triangle by definition of triangle? and how about if it has 2 sides, if not 3 sides are zero?
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. Find the projection of the triangle on the coordinate planes.

Given the following, three vectors: a⃗ =3i−2j+5k b⃗ =i−6j+6k c⃗ =2i+3j−k Relative to cartesian coordinate systems with origin O. I calculated the sides to be 4.58,11.45 and 7.87. I also calculated ...
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Triangle inscribed in another triangle

If $a,b,c$ are the sides of a triangle,$\lambda a,\lambda b,\lambda c$ the sides of a similar triangle inscribed in the former and $\theta$ the angle between the sides $a$ and $\lambda c$,prove that ...
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Is my proof of 'inscribed angle theorem' different from the usual one?

The usual proof of inscribed circle theorem uses the fact that the sum of interior angles is equal to exterior angle. I've formulated a little different approach, although the underlying concept is ...
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Area of equilateral triangle from circumcircle

I am trying to calculate skewness of triangle. Given the sides of a triangle (not equilateral), I calculated circumradius from which I would like to get area of equilateral triangle.
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Given verticies find the area of the triangle formed

When I looked at this problem I didn't think it seemed all that hard until I actually tried it. The problem is this: Given the rectangular vertices $O(0, 0, 0), P(-1, 2, -3), Q(-2, 3, -4), R(0, 0, ...
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315 views

Energy of a Triangle Wave

I want to find the energy of two triangular functions (identical, one above ( S1(t) ) and the other below ( S2(t) ) the x axis, so it should be the same thing). They are shown in the images below. The ...
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Angle for sloped surface intersection

I'm not sure if this is the right SE site for this questions so apologies if it's not. But I've got a question about calculating angles that I can't seem to figure out the maths for. To add a bit of ...
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Ideal Triangles and Klein Beltrami Disc

I'm trying to prove something with the ideal triangle in hyperbolic geometry and someone told me that the ideal triangle looks like a euclidean triangle inscribed in a circle in the Klein Beltrami ...