For questions about properties and applications of triangles

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1answer
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How to maximize the function

I have a triangle $T=ABC$. I want to calculate $\max (a-b)$, where the the angle $ABC = \beta$, and $|AB|=c$ is fixed (pre-known). My guess is $c\times\cos (\beta)$, but I want to prove it. Let ...
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2answers
729 views

How to calculate the angle between two vectors, defined by 3 points on the earth?

I want to develop a formula to calculate the angle between two vectors. The vectors will be OX and OY (from point O to X , and Y), where the points are defined by their latitude and longitude values. ...
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1answer
325 views

What is a generic triangle?

I think the question speaks for itself. I came across this term in one exercise, but am not sure what it is. The definition my textbook gives is a triangle where the three vertices are free. I am ...
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2answers
79 views

Proof using properties of an isosceles or right-angle triangle

Given a triangle $ABC$ with sides $AB=BC$ and angle$\angle B=100^\circ $, prove that $$a^3 + b^3 = 3a^2b$$ where $a=AB=BC$ and $b=AC$, I have tried to use simultaneously the sine and cosine rules as ...
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1answer
4k views

Find height of a triangle given length of three sides?

How can I find the height of a triangle given the length of all three sides? The only solution I could find was to use Heron's formula to find area then $A=\frac{1}{2}bh$ to find height. Is there an ...
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0answers
85 views

Minimize the perimiter of a triangle with an inscribed circle

A circle touches the two legs of an angle. How can one draw a line that intersects both legs, such that the circle lies within the triangle with as sides the two legs and the drawn line, and such that ...
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0answers
66 views

Probability of a triangle in a circle [duplicate]

I'm confused on my calculations on analytic geometry with probability. Things I learned on these were messed up since I was a newbie on these subjects. Here's my problem: Three points are chosen ...
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1answer
111 views

Inequality in triangle

Let $ABC$ be a triangle and $M$ a point on side $BC$. Denote $\alpha=\angle BAM$, $\beta=\angle CAM$. Is the following inequality true? $$\sin \alpha \cdot (AM-AC)+\sin \beta \cdot (AM-AB) \leq 0.$$
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2answers
121 views

Finding the measurement of an angle

I have been stumped on this problem for a couple days now, and I would like some help solving it. Here is the picture that I drew up: ABCD is a regular square. Line FG is a perpendicular bisector ...
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2answers
3k views

What is the maximum area of a square inscribed in an equilateral triangle?

What is the maximum area of a square inscribed in an equilateral triangle? Please post the approach to solve the above question.
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1answer
143 views

Prove, square of quadrilateral is the sum of squares of 4 triangles [duplicate]

Let A1, B1, C1, and D1 - midpoints of the sides AB, BC, CD and DA convex quadrilateral AВСD. Directs AC1, ВD1, CA1 and DВ1 - divide it by 5 quadrilaterals and 4 triangles. Prove that the area of ​​the ...
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Determine if projection of 3D point onto plane is within a triangle

In 3D, given three points $P_1$, $P_2$, and $P_3$ spanning a non-degenerate triangle $T$. How to determine if the projection of a point $P$ onto the plane of $T$ lies within $T$?
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Is there any equation for triangle?

Like there's an equation of a circle, is there any equation of a triangle? I've been trying to build one and the closest thing I've managed to do is to create an equation of 2 lines and use the $x$ ...
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2answers
380 views

In an equilateral triangle what is sum of distance from vertices to a point inside the triangle?

In an equilateral triangle what is sum of distance from vertices to any arbitrary point inside the triangle? What is the relation between $a$ and $x + y +z$. The special condition is that the ...
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0answers
42 views

Trigonometry: Isosceles Triangle [duplicate]

I saw the following problem on Facebook (figure not drawn to scale): ...
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5answers
123 views

Right Triangles and Altitudes

I am once again stuck on a question about geometry, this problem is about altitudes that crate right triangles: Let there be a triangle that has side lengths of 13, 20, and 21. Given this, find the ...
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1answer
74 views

Finding inradius given the heights

I'm given the heights of a triangle. Find the inradius. I know that inradius is area/semiperimeter. But then?
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3answers
1k views

Proving the length of angle bisector

How do I prove that a triangle with sides a, b, c, has an angle bisector (bisecting angle A) is of length: $$\frac{2 \sqrt{bcs(s-a)}}{b+c}$$ I have tried using the sine and cosine rule but have ...
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1answer
549 views

Triangle Point Picking in 3D

To take random uniform points inside a triangle Triangle Point Picking method is used. But this is for 2D points, how can I take random points from a triangle that is defined by 3 arbitrary 3D points? ...
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1answer
48 views

Formula for the length of line that connects two sides of a triangle.

For the triangle in the picture, coordinates of $A$, $B$ and $C$ are known. Is there an explicit formula for length $XY$, as a function of height $h$? It's a function of other variables as well, but ...
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1answer
73 views

About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true? "Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of ...
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1answer
2k views

Related rates: Find dA/dt of triangle, given d(theta)/dt — Can't come to textbook answer

The question: ABC is a triangle in which the lines $\overline {AB} = 20cm$, $\overline {AC} = 32cm$ and $\angle BAC = \theta$. If $\theta$ is increasing at the rate of 2° per minute, determine the ...
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2answers
328 views

Some angles of triangle inscribed in circle and intersection of its bisectors with that circle

A triangle $ABC$ has been inscribed in a circle. The bisectors of angles $A$, $B$ and $C$ meet the circle at $P$, $Q$ and $R$ respectively. If angle $BAC = 50^\circ$, then what would be the value of ...
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3answers
410 views

A triangle has side lengths 4,6,8. A tangent is drawn to incircle parallel to side 4 cutting …

Problem : A triangle has side lengths $4,6,8$. A tangent is drawn to incircle parallel to side $4$ cutting other two sides at M and N, than length of MN is (a) $\frac{10}{9}$ (b) $\frac{20}{ 9}$ ...
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2answers
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Maximum area of a square in a triangle

I want to calculate the area of the largest square which can be inscribed in a triangle of sides a, b, c. The "square" which I will refer to, from now on, has all its four vertices on the sides of the ...
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3answers
76 views

Trigonometric problem in triangles.

I need your help. I'm studying physics, but I have a trigonometric problem. I attached a figure where depicts the angles and the unknown $x$. The idea that I want to understand is how to express $x$ ...
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4answers
665 views

What is so special about triangles?!

Take any random triangle. If we draw internal-angle-bisectors of all its angles, they intersect at the same point. If we draw the perpendicular bisectors of each side (although they aren't ...
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1answer
896 views
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1answer
127 views

Triangles within square

Points E and F lie on the sides BC and CD of rectangle ABCD, the AEF is an equilateral triangle. point M is the midpoint of the AF. Prove that the triangle BCM is equilateral.
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1answer
67 views

(Non-)Uniqueness of a given triangle

Let's assume that some triangle is described by its two sides and an angle (which is not between the given two sides however). Basically for a triangle above only characteristics $c,\ b,\ C$ are ...
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2answers
410 views

Rectangle divided into three triangles with two lines. One angle is given, what are all the others?

Let's suppose I have a rectangle divided into three triangles in the following way. No lengths of either the rectangle or triangles are known, only one angle is known. I would like to know how to ...
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1answer
111 views

Correct my Pre-Calculus work please?

PPlease explain how to do these problems. I got my test back and I'm trying to see what I did wrong so I can do better on the next test. Given $g(x)=\sqrt{x+5}$ find $g^{-1}$ $x= \sqrt{y+5}$, I ...
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2answers
130 views

When does the triangle have the smallest area?

The following triangle has an area $S$, and the sides $AO$ and $BO$ have the length $a$ and $b$, respectively. There is a fixed point $X$ at $(x,y)$. A point $C$ is put on the line segment $OA$, and ...
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2answers
122 views

Triangle and its circumcircle

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. On sides $AC$, $BC$ lies respectively points $E$, $F$ such that $CE=BE$ and $CF=AF$. $CM$ is a median of triangle $EFC$. Show that line $CM$ pass ...
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1answer
100 views

Good websites/books for geometry exercises?

I'm looking for exercises similar to those seen on putnam exams or olympiad exams, such as finding the area of polygons inscribed other polygons, finding certain angles, etc.
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1answer
23 views

Simple Triangle Completion

How do you find the missing point of a triangle, given: 2/3 of the points, two slopes, and one angle of direction. Here's the problem. There are two points: Point B (1,1), showing an arrow going ...
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2answers
610 views

Use the law of cosines to derive the triangle inequality

I am given the vectors: and show that they span the triangle with sides $a,b,c$ with $c=||u-v||$ and determine for which $\gamma∈[0, \pi]$ we have equality. Any help is appreciated.
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4answers
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finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in x-y plane? One approach is to find the length of each side from the coordinates given ...
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2answers
934 views

Find an angle of an isosceles triangle

$\triangle ABC$ is an isosceles triangle such that $AB=AC$ and $\angle BAC$=$20^\circ$. And a point D is on $\overline{AC}$ so that AD=BC, , How to find $\angle{DBC}$? I could not get how to use ...
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1answer
105 views

Getting an angle

I have a unit circle, and two angles: $\alpha=\angle{JON}\in[0,\pi]$ and $\beta=\angle{IOM}\in[0,\frac{\pi}{2}]$. Using angles, we can get points $N$, $M$ as on the image. Then, dropping a ...
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1answer
70 views

About the area of integer-edge-length triangles

Let $a,b,c$ be three edge lengths of a triangle whose area is $S$. Then, here is my question. Question : Supposing that $a,b,c$ are natural numbers, then does there exists $(a,b,c)$ such that ...
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1answer
127 views

Triangle problem (only know 1 side and y-axis coordinates of 2 points)

I have a triangle problem that I am in desperate need of help with with. Here is what I know ... the triangle is mapped on a graph where I only know y-axis coordinates for 2 points, the length of one ...
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2answers
99 views

Generating Pythagorean Triples S.T. $b = a+1$

I am looking for a method to generate Pythagorean Triples $(a,b,c)$. There are many methods listed on Wikipedia but I have a unique constraint that I can't seem to integrate into any of the listed ...
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1answer
75 views

non right angle triangle - solve for B when b, A and a are known

I'm trying to work out the two possible values for B when A, a and b are all known. I'm certain its possible but I'm not sure how to start or what theories to look for to solve the question as my math ...
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2answers
865 views

How to find sum of 3 perpendiculars of a triangle?

Q. ABC is an equilateral triangle with side 10cm and P is a point inside the triangle, at a distance of 2cm from AB. If PD, PE and PF are perpendiculars to the three sides, find sum PD+ PF+PE. ...
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1answer
146 views

Lines $ MF, DE, QR$ in a triangle intersect at one point

In a triangle ABC, a circle is inscribed with center in $I$. The inscribed circle touches sides $BC,CA,AB$ in $D,E,F$ respectively. Join the point $C$ and $F$, $B$ and $E$. Let $Q$ and $R$ be the ...
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4answers
469 views

How do you find the area of a triangle in a 3D graph?

How do you find the area of a triangle in a 3 dimensional graph? Is it any different than a regular 2d graph? How would you solve it, if these were your three points? A(1,-4,-2), B(3,-3,-3), ...
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4answers
144 views

Is there an integer that $\sqrt{3}$ can be multiplied by that will produce a whole integer?

The question came up while messing around with graph paper. I wanted to make an isosceles triangle where the length of one side and it's hight were both integers. The closest I could get was a base ...
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2answers
2k views

How to find the type of triangle when given the ratio of it's sides?

Q.The sides of a triangle are in ratio 4 : 6 : 7, then the triangle is: (A) acute angled (B) obtuse angled (C) right angled (D) impossible It's definitely not (C) right-angled ...
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1answer
122 views

A question about a very peculiar triangle.

If we have a triangle where the Perimeter >0 and the Area >0 , and Area=Perimeter, what special condition must the angles of this triangle satisfy for this to happen? I've done a bit of research and ...