For questions about properties and applications of triangles

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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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3answers
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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 ...
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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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1answer
71 views

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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45 views

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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103 views

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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2answers
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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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3answers
116 views

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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2answers
29 views

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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0answers
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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 ...
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1answer
36 views

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A.

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A. I know it looks simple, but my reasoning does not agree ...
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Trying to prove concurrence of altitudes of a triangle.

I know that this question had been asked before, but I am not exactly following what the answers say. Doing my own way here: I am puuzzled how to continue? I named the points A,B,C, and the foot of ...
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1answer
49 views

who discovered the orthocenter of a triangle?

I tried to answer Is there a name for this result in planar geometry? and wanted to go back to the first mention of the orthocenter (or even the altitude of a triangle, but i did draw a complete ...
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What's wrong with my reasoning while setting up a limit?

I was writing an answer to this question, which asks about what happens to the apex of an isosceles triangle if a vertex is at infinity. I thought it would be very easy to prove it by setting up a ...
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171 views

Triangle-free graph with 5 vertices

What is the maximum number of edges in a triangle-free graph on 5 vertices? No answers, please...just hints. I believe that E $\leq$ 5, but I'm not sure where to go from there.
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Angle in a triangle within a circle.

A and B are two points on the circumference of a circle with centre O. C is a point on OB such that AC $\perp OB$. AC = 12 cm. BC = 5 cm. Calculate the size of $\angle AOB$, marked $\theta$ on the ...
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1answer
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Show that the midpoint of $AB$, $AC$, and $DE$ are aligned.

Let $ABC$ be a rod, $D$ and $E$ two points such as: $\vec{EC} = k \cdot \vec{EA} / \vec{DA} = k \cdot\vec{DB}$. How can I show that the midpoint of $AB$, $AC$, and $DE$ are aligned?
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Lemoine Point triangle

from Wolfram MathWorld, I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum. How can I ...
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4answers
55 views

In triangle ABC, Find $\tan(A)$.

In triangle ABC, if $(b+c)^2=a^2+16\triangle$, then find $\tan(A)$ . Where $\triangle$ is the area and a, b , c are the sides of the triangle. $\implies b^2+c^2-a^2=16\triangle-2bc$ In ...
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Intesection point of feet of altitudes

If triangle has vertexes at $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, is the intersection points of feet of altitudes $$x_h = \frac{x_1x_2(y_2-y_1) + x_2x_3(y_3-y_2) + x_3x_1(y_1-y_3) + y_1^2(y_3-y_2) + ...
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1answer
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Finding the minimum value of squares of sides of a quadrilateral

What is the minimum value of $\frac{a^2+b^2+c^2}{d^2}$ where $a,b,c,d$ are the sides of quadrilateral I assumed the diagonals to be $p$ and $q$. I got that for minimum angle $A$ and $C$ must be ...
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2answers
39 views

Length of a right triangle created by skewing a rectangle's edge by a fixed amount

I have the above problem for a grid-based graphics system I'm working on, and I'm not sure if the math is solvable or not. I'm trying to determine the value of $A$. I've attempted to use ...
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1answer
66 views

Finding the value of k

If $x,y,z$ are perpendicular distances from circumcenter on the sides $BC,AC$ and $AB$ respectively. In need find $k$ such that $$\frac ax+\frac by+\frac cz=\frac{abc}{kxyz}$$ (Lowercase letters ...
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Finding the third side of a triangle, given ratio of two sides and difference of two angles [closed]

Given $a=2b$ and $|\angle A-\angle B|=60$ degrees. Find the third side, where lowercase letters denote opposite sides and uppercase letter angles. Progress I could find the $\cos C$ but then ...
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Prove that $\frac{1}{2}ab \equiv \int_0^b \! f(x) \, \mathrm{d}x$ when calculating the area of a right triangle.

Triangle $ABC$ is a right triangle with sides $AB$, $BC$ and $AC$. $a$ is the length of $AB$. $b$ is the length of $BC$. $c$ is the length of $AC$. If $a = 3$, and $b = 4$, we can use ...
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Sum of segments inside a right triangle.

I am interested for a problem involving the sum of segments inside a right triangle. Consider a right triangle of hypotenuse $\overline{BC}$ and catheti $\overline{AB}$ and $\overline{AC}$. From the ...
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1answer
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Geometry Problem about tangent lines

Let S be the circumcenter of ABC. $A_0$ is the middle of arc BC not containing A, $C_0$ also the middle of arc AB without C. Let $S_1$ be a circle with center $A_0$, tangent to BC, $S_2$ with center ...
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How to find mass points and ratios in a triangle?

How to find mass points with weights and ratios is my question. In my class, we learned about mass points. First we had the given ratios of 2 side lengths. Given: MC = d MB = e MA = f BD:DA = ...
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An altitude is divided into 5 equal parts by 4 lines. Prove that the the areas of alternate sections are equal.

The question is as follows : Let their be a triangle ABC. Make altitude AD on C. Divide this altitude in 5 equal parts with lines EF, GM, IJ, KL intersecting at points M,N,O,P respectively. We have ...
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Trying to figure out coordinates of isoscleles triangle

I'm programming some application, which heavily depends of geometry. Let's say, in 2D coordinates system I have i.e. : Bxy = (5,-2) Cxy = (2,-5) ABlength = 5.5 ...
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In a Right Angled Triangle.

In a triangle ABC, Let $\angle$C=$\frac{\pi}{2}$. If $r$ is the inradius and $R$ is the circumradius, then what is the value of $2r+R$. Options are a+b b+c c+a a+b+c My approach. Radius of ...
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How is the area of this triangle calculated

I was reading "Problems of Calculus in one variable" by I A MARON, and came across this solved example in first chapter which I am unable to comprehend, please help me understand this. Scan of the ...
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Find cosine of acute angles in a right triangle.

If sides of a right triangle are in Geometric Progression, then find the cosines of acute angles of the triangle. [Answer] $\frac{\sqrt{5}-1}{2}$,$\sqrt\frac{\sqrt{5}-1}{2}$ My work: Using ...
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Two questions regarding the angle of reflection

I have two problems regarding the calculation of angles given certain values. In the first problem I need to calculate the angle X given that both angles Y are identical In the second problem I ...
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2answers
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North has 0 degree and right angle has 90 degree although both are in same position

Before reading trigonometry I guessed that if a line is pointing to north then it has 0 degrees and increases clockwise. But now I see right angle has 90 degree though that is in the same position as ...
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How to find circumference origin position?

I need to find origin of circumference which is defined by two points and vertex angle of isosceles triangle: I've got radius of triangle by ...
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How was this equation for the hypotenuse of a triangle derived?

I've been staring at this for quite a while and simply can't understand how they got the equation for the hypotenuse. Probably has something to do with it being 5am my time! I'm confused because ...
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Finding the ratio of division by circumcenter

In an acute angled triangle ABC where O is the circumcenter Prove that $BD : DC = sin2C:sin2B \quad$ where D is the point of intersection of AO (extended) with BC. $AO : OD = sin2C + sin2B : ...
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using slope to find an angle in right angled triangle

I have a right angled triangle in which I know the length and the slope of the hypotenuse, how do I find one of the angles? Thanks
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$1,2,3$ task — calculate tangents

Given is square $ABCD$. Point $E$ is the midpoint of segment $CD$ ($E\in CD \wedge |DE|=|EC|$). Point $P$ is common point of diagonal $AC$ and line segment $BE$. ($\lbrace P\rbrace = AC \cap BE$). ...
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Coordinates of a vertex of a triangle?

Here is the problem: There is a triangle with vertices $A,B,C$ in a cartesian coordinate system, where coordinates of points $A$ and $B$ and the angle $\alpha=\measuredangle ABC$ are given. The ratio ...
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How many triangles in the picture?

Sorry if this has already been asked before. Is there any formula for such questions? EDIT: I have numbered the smallest triangles in the picture and marked the pentagon as x. Then I listed all ...
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2answers
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$A,B,C$ satisfy $\sin 2A: \sin 2B: \sin 2C= 5:12:13$ find $A$?

I would appreciate if somebody could help me with the following problem: Question: $A,B,C$ satisfy (1), (2) (1). $A+B+C=\pi(0< A,B,C< \pi)$ (2). $\sin 2A: \sin 2B: \sin 2C= 5:12:13$ Find $A$ ...
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Find length of side

I tried to solve this problem ... but i can't find answer. Anyone can help me? EBC=90 & DCB=90 & AHC=AHB=90
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Prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$

If $x, y, z$ are the sides of a triangle, then prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$ This is a self-answered question.