For questions about properties and applications of triangles

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0answers
20 views

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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1answer
24 views

Finding other two vertices of a triangle from centroid and one vertex?

I am working on some gesture recognition for my game and I want to find if a point is inside the triangle created by the user or not. For that I need three vertices. Currently I am using the '$1 ...
1
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1answer
95 views

Find closest point in triangle given barycentric coordinates outside

Given a non-degenerate triangle ABC and an arbitrary point P in 3D space, I can project P onto the plane defined by ABC and check whether the triangle contains it as described here. I end up with ...
2
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1answer
49 views

Minimum Distance between a Triangle and a Distance Field 3D

I am looking for (possibly numerical) solution to this geometric problem: Given a filled 3D triangle $T = \text{conv}(p_1, p_2, p_3) \subseteq R^3$, and a distance field $D(x) : R^3 \to R$, what ...
10
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1answer
101 views

Is there a name for the recursive incenter of the contact triangle?

Recently, I became aware that there are many more triangle centers than the four I learned about in school. This reminded me of a thought I had when I first learned about the incenter: what point ...
3
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1answer
76 views

Closest point on a 3D triangle, is this algorithm correct?

Given a point $P$ and three triangle vertices $U$, $V$, $W$, all in $\mathbb{R}^3$, I need to find the point in the triangle $UVW$closest to $P$. Does the following algorithm work, or have I missed ...
3
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2answers
109 views

Why is the volume one third of that? I mean, where's the fault in my logic? [duplicate]

The volume of a cuboid is $l \times b \times h$. That is, it is equal to base area times height. I think it means that the base is added up height times, it becomes volume (It makes sense if we think ...
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3answers
98 views

Coordinate of the excentre of a triangle

I am just wondering that how the coordinate of the excentre comes out if we know the coordinates of vertices of the triangle.
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2answers
42 views

Proving that the orthocenter lies on $OD$?

While trying to solve this question using GeoGebra, I realized the following curious thing: If $I$ is the incenter of $\triangle ABC$, $ID \perp BC$ with $D$ on $BC$, $AD \perp IO$ with $O$ on ...
3
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2answers
65 views

Find the angle if the area of the two triangles are equal?

Let $I$ be the incenter of $\triangle ABC$, and $D$, $E$ be the midpoints of $AB$, $AC$ respectively. If $DI$ meets $AC$ at $H$ and $EI$ meets $AB$ at $G$, then find $\angle A$ if the areas of ...
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3answers
55 views

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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1answer
80 views

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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1answer
27 views

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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2answers
45 views

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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2answers
65 views

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
49 views

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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5answers
164 views

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 ...
16
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1answer
221 views

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
74 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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0answers
49 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?
2
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0answers
54 views

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 ...
3
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1answer
106 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
85 views

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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3answers
158 views

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
118 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
30 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 ...
2
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0answers
76 views

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 ...
4
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2answers
113 views

Three circles having centres on the three sides of a triangle

NOTE: I would appreciate it if you provided a hint and not the whole solution. BdMO 2014 Nationals: In acute angled triangle ABC, considering a portion of side BC as diameter a circle is drawn ...
0
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1answer
39 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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1answer
65 views

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
65 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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0answers
40 views

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 ...
4
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4answers
206 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.
5
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3answers
143 views

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 ...
2
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0answers
31 views

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 ...
0
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4answers
60 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 ...
2
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0answers
44 views

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) + ...
0
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1answer
21 views

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 ...
0
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2answers
43 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 ...
0
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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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1answer
46 views

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 ...
0
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2answers
54 views

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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3answers
23 views

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 ...
2
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1answer
59 views

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 ...
1
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0answers
98 views

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 = ...
1
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1answer
34 views

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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1answer
23 views

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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1answer
63 views

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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1answer
32 views

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 ...
2
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1answer
42 views

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 ...