# Tagged Questions

For questions about properties and applications of triangles

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### Proving $\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C$

As the title, By considering $\bigtriangleup$ABC, Prove $$\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C$$ Thanks
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### 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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### Proving that $AB = AC$. [closed]

In a $\triangle ABC$, $D$ is a point on $BC$ such that $AB+BD=AC+CD$.Let the centroids of $\triangle$s $ABD$ and $ACD$, vertices $B$ and $C$ lie on a circle.Prove that $AB = AC$.
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### Find point in right triangle with given one vector and one point

I am developing a game where the user move a car with his finger. The car is represented as vector (one point and angle of rotation in the screen). When the user start to dragging the car he generate ...
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### Prove that $\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}$ for angles of a triangle

Let the three internal angles of a triangle are $a,b,c$. Prove that $$\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}.$$ I'm looking for an elementary, geometric proof. So avoid derivatives and ...
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### Nine-point circle equivalent for tetrahedrons?

Nine-point circle for a triangle is defined as the circle that passes through: the midpoint of each side the foot of each altitude the midpoint of the line segment from each vertex to the ...
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### Proving $HH', BB', CC'$ are concurrent?

The orthocenter of $\triangle ABC$ is $H$. Let $B'$ be a point on AC and $C'$ be a point on AB, such that $BCB'C'$ is a cyclic quadrilateral. Let the orthocenter of $\triangle AB'C'$ be $H'$. ...
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### ABC is a triangle, D is a point in the triangle. E is the midpoint of BD. AB=BC, angle ABD= angle DBC=35 degrees, angle ACD=25 degrees. Angle BAE=?

I tried to solve this problem but couldn't. I just know that here, angle BDC= 100 degrees, angle BAC= 40 degrees, AB^2+AD^2=2(AE^2+BE^2) and AB/AD={sin(angle DAE)}/{sin(angle BAE)}
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### Intersections of convex hulls

Given a set of $n$ points $\{A_1, \ldots , A_n\}$ of the plane and every possible triangle formed with $3$ points $A$, I would like to describe the intersections fo theses triangles. By intersection, ...
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### Finding other two vertices when one vertex and each point on the triangle is known ?

I am working on some gesture recognition for my game. I am stuck on a problem. I have one vertex i.e the starting point and every point on the triangle, I also have the centroid. So how do I find the ...
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### Sum with three notations around it.

Have seen the below notation (How to calculate number of triangles and points after dividing a triangle n times?) and need to break it down into plain english so to speak. This just so I can catch up. ...
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### How to calculate number of triangles and points after dividing a triangle n times?

When having a triangle and dividing it n times, how to get the number of triangles and points? ...
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### Solution of triangles

Find the angle at the vertex of an isosceles triangle of maximum area for the given length 'l' of the median to one of its equal sides. I tried to get a relation between l and one of the equal sides ...
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### Finding the area of a triangle, given the distance between center of incircle and circumscribed circle

Consider the following depiction: $ABC$ is an isosceles triangle ($AB=AC$), where the two angles opposite the equal sides are equal $\beta$ ($\beta>60$), and $AD$ perpendicular to $BC$. $O$ is ...
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### Minimise the Sum of the Areas of the Circumcircles

In a triangle $ABC$, point $X$ is picked on $BC$ such that the sum of the areas of the circumcircles of $ABX$ and $ACX$ is minimised. Describe where $X$ would be located on $BC$, and prove that this ...
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### How to determine the equation and length of this curve consistently formed by the intersection of Circles

Consider a Point $A$ that moves linearly on the positive $x$-axis with the velocity $1$ m/s and another Point $B$ at a distance $L$ from $A$ with position $(L,0)$. With each forward motion of point $A$...
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### Area of triangle with given coordinates of the vertices

The question for my math is: "Sharon made a scale drawing of a triangular park. The coordinate for the vertices of the park are: $(-10,5)$, $(15,5)$, $(10,12)$. What is the area of the triangular ...
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### Under what conditions will the rectangle of the Japanese theorem be a square?

In geometry, the Japanese theorem for cyclic quadrilaterals states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle. Question. Under ...
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### Root of sum of squared distances

Say I want to calculate the euclidean distance of all edges of a triangle. I could take the root of the squared distance of each edge and add those. This would give me the right result. Adding up ...
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### Exact values on unit circle

Why is it allowed to draw an equilateral triangle on the unit circle to prove the exact values for $\cos(\pi/3)$ or $\sin(\pi/3)$ for example?
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### How prove that $II^{\prime}< AA^{\prime}$ for $I$ and $I^{\prime}$ be their incenters?

Assume that we have two triangles $ABC$ and $A^{\prime}BC$. Let $I$ and $I^{\prime}$ be their incenters. How prove that $II^{\prime}< AA^{\prime}$? I have no idea how to do this, can this be ...
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### Inequality of length of side of triangle

For any triangle with sides a,b,c $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$$ I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn't help in any sense except wasting 3 pages that lead to ...
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### How to find the inradius of a triangle with given side lengths?

I need to find the inradius of a triangle with side lengths of $20$, $26$, and $24$. I know the semiperimeter is $35$, but how do I find the area without knowing the height? Thank you.
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### How can I find the sine or the tan or the cos of an angle in radian?

There is an angle equal 0.54 radians and opposite leg equal to 3 units, I need to find the length of the adjacent leg. I know that I have to do ${\rm leg} = \frac{3}{\tan(0.54 \text{ rad})}$. I got ...
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### The area of intersection of an isosceles triangle with another triangle

I tried graphing the equations that form the two isosceles triangles and integrating the bounded area and got 7.456 as my answer after rounding. The answer key has the answer listed as 7.2 However, ...
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### Where to put angle ending on right triangle, only using variables.

Let's say I have a triangle ABC, with side lengths abc. I need to draw a line from the angle connecting the base (c) and hypotenuse (b). I don't know the real angle, but I know it's sin-1. I need to ...
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### Looking for an existing proof for a property of triangles

In my paper, I need the following lemma. I can prove it, but it is a little lengthy to be put inside the paper. I am wondering is there any existing proof that I can quote. Lemma 1: Let the nodes ...