For questions about properties and applications of triangles

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10
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1answer
108 views
+50

A curious triangle inequality

Let $ABC$ be a triangle. Pick a point $P$ inside the triangle. How would you show that \begin{equation} |PA|+|PB|+|PC|+\min\{|PA|,|PB|,|PC|\}\leq |AB|+|BC|+|CA|. \end{equation}
0
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0answers
25 views

Find $\angle B$ if $AD=\frac{abc}{b^2-c^2}$

If AD is median and $AD=\frac{abc}{b^2-c^2}$ $[b>c]$ and $\angle C=23^{\circ} $. Find $\angle B$ Is this information sufficient to find $\angle B$? I tried using sine rule in triangle $ADC$ and ...
1
vote
1answer
17 views

prove that $MN \parallel BC$ in an equilateral triangle

$\Delta ABC$ is equilateral with $M$ and $N$ being interior points. if $\angle MAB=\angle MBA=40^{\circ}$ $\angle NAB=20^{\circ}$ and $\angle NBA=30^{\circ}$. Prove that $MN \parallel BC$ from ...
-2
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0answers
35 views

how to get length of side of right triangle [on hold]

Please help in determining the length of the sides of the triangle marked [?].
1
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2answers
38 views

Triangles - sin, cos etc. [on hold]

I know this is a quite simple question for most of you out there. However it has been a little troubling for me, and would like to get a little help if possible. I have a triangle $ABC$ where I know ...
0
votes
2answers
38 views

What trigonometric identity makes the method of triangulation work?

I've read the article on Wikipedia, but I don't get how to construct the relationships between sides and angles to reach a solution for the distance between two points. All the other sites I read ...
0
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1answer
21 views
0
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0answers
19 views

Prove that the area of a triangle DEF is correct.

There's any triangle ABC. First player 1 has to set D on AB so that in the end the triangle DEF has the highest possible area. Second player 2 has to set E on BC so that in the end the triangle has ...
0
votes
1answer
33 views

Find the sides of a right triangle formed by connecting two other right triangles from the center of their hypotenuse.

I have the following sketch of the problem: I need to find the values of $x$ and $y$ in the previous drawing. The hypotenuses of both black triangles are of equal length and the red triangle is a ...
0
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0answers
41 views

Synthetic geometry , angles.I need some ideas

Let $ABC$ be a triangle such that $m(\measuredangle ACB)>30$ and $M$ in the interior of the triangle with $m(\measuredangle BMA)=120, m(\measuredangle BCM)=30$. Let$\{ D\} = AM\cap BC$ and $P \in ...
2
votes
1answer
27 views

Proof of Cauchy-Schwarz Inequality 1

In my lecture notes I've written the proof of Cauchy-Schwarz inequality as: Let t $\in$ R and $\langle x+ty, x+ty\rangle \geq 0$, then $\langle x+ty, x+ty\rangle $ = $\langle x, x+ty \rangle + ...
0
votes
2answers
24 views

A Quadrilateral and A Triangle in a Trapizium

In the above diagram, $ABCD$ is a Trapizium with $AD || BC$ and $BC \perp AB$ $AB = 20, \; AD = 6,\; BC = 30$ $M$ is a point on $DC$ such that $[ADMB] = [BMC]$, where $[x]$ denotes the area of $x$. ...
1
vote
2answers
60 views

Integrating triangle in a 2D plane

I am interested in integrating $(x^2y+y^2x)$ on the following loop: $(x=1,y=2)\rightarrow(x=2,y=1)\rightarrow(x=3,y=3)\rightarrow(x=1,y=2)$. I know this loop forms a triangle with all three sides ...
0
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0answers
18 views

Using oblique projection can you always rotate a triangle to look like an equilateral triangle? [duplicate]

Starting with any triangle using oblique projection, can you view any shape triangle from an angle to see it as an equilateral triangle?
0
votes
3answers
49 views

How do I compute the angles of a pyramid from the angle between its sides?

I have been given the following problem to solve: In a right pyramid whose base is an equilateral triangle, the angle between 2 side-faces is 70 degrees. Compute the base angle of a side-face. I ...
1
vote
3answers
44 views

Distance of centroid to incenter

Suppose there is a right triangle where all side-lengths are integers. The distance from the circumcenter to the centroid of the triangle is 6.5. Find the distance from the centroid to the incenter ...
-2
votes
0answers
16 views

Angle in triangle [closed]

We have triangle ABC with point E on side AB and D on side BC. We know that AE=AD and BD=AC. Is it true that angle BAD is equal to ABC and if yes, how to show it?
2
votes
3answers
30 views

New coordinates after clockwise rotation of triangle?

The figure below represents a triangle $PQR$ with initial coordinates of the vertices as $P(1,3)$, $Q(4,5)$ and $R(5,3.5)$. The triangle is rotated in the $X-Y$ plane about the vertex $P$ by angle ...
0
votes
1answer
34 views

Find $|CM|$, if $|CA|=a$ and $|CB|=b$. [closed]

Let $O$ be a center of a circle, circumscribed over $\triangle ABC$. Perpendicular, drown from the point $A$ on the line $CO$, cross the line $CB$ in the point $M$. Find $|CM|$, if $|CA|=a$ and ...
0
votes
1answer
361 views

Finding Y coordinate of third triangle point when X coordinate and two other points are already known

Suppose you know the coordinates for points A and B of a triangle. We can refer to those coordinates as (Ay,Ax) and (By,Bx). Also, suppose you know the X coordinate for point C (Cx) but do not know ...
1
vote
1answer
34 views

Prove that $MN = \dfrac{|b − c|}{2}$

In triangle $ABC$, point $M$ is the midpoint of $BC$ and $N$ is on the angle bisector of $\angle A$ such that $MN \parallel AB$. Prove that $MN = \dfrac{|b − c|}{2}$. Attempt: I drew it out and ...
0
votes
1answer
58 views

Prove for a area relationship in a pedal triangle

Let $\triangle ABC$ an acute triangle and call $K,L, M$ the orthogonal projections of $A,C$ and $B$ on the opposing sides. Prove: $A_{\triangle KLM} = 2 A_{\triangle ABC}\cdot \cos \hat A ...
1
vote
0answers
44 views

In a triangle with sides $a, b,c$ and the relation $a^3=b^3+c^3$, get the angle between $b$ and $c$. [closed]

In a triangle with sides $a, b,c$ and the relation $a^3=b^3+c^3$, get the angle between $b$ and $c$.
0
votes
2answers
42 views

Geometry/ Triangles problem

I have been struggling with this problem, and I think it should be possible to solve but right now I cannot find how. Given two coordinates/points (x1,y1) and (x2,y2) The angle d1 with the ...
0
votes
3answers
36 views

Find the ratios of the sides of a triangle

If the perimeter of a the right-angle triangle is six times its smallest side, find the ratios of the three sides. I tried solving it by using the normal area and volume.
2
votes
1answer
371 views

Geometry - optimal 2D mesh between X expendable points

Say you have X points on a plane. If you connect two points, you form a line. Connecting three points forms a triangle. A line cannot cross a line, and a smaller triangle cannot be created inside a ...
1
vote
1answer
31 views

Proof concerning isosceles triangles

In the triangle $ABC$ it is $AC = BC$ and $\alpha = \beta$. The points $D$ and $E$ are on the line through $A$ and $B$. Show that the triangle $CDE$ is isosceles. Hey there! Is it sufficient ...
0
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0answers
41 views

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?
0
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1answer
24 views

Proof with area segments in a triangle

I have to show that $A M_CS$ and $M_CBS$ have the same area $X$ and that concerning areas $X=Y=Z$ is true. I'm really stuck here, I would appreciate any help or tip...! How can I start here?
0
votes
3answers
50 views

Prove a parallelogram inside parallelogram

I have drawn a figure, In parallelogram ABCD, AP is the bisector of angle A CQ is the bisector of angle C Can I prove APCQ is a parallelogram? or it isn't? I first joined AC and now if somehow I ...
0
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2answers
38 views

I need some help with Geometry. Is this a correct answer to this problem?

Good day, I have a question regarding geometry. I don't know whether my answer is correct because the answer in my book uses a totally different method for solving this particular problem. Here's ...
-2
votes
2answers
57 views

is there a triangle with sides $2,3,6$?

Is there a triangle with $a=2, b=3, c=6$? (I know there's not because sum of any two sides has to be greater than the third side) How much do we need to extend these sides to get a right triangle ...
0
votes
1answer
31 views

Find sides of a right triangle given hypotenuse c and area A (no numbers given)

I've solved couple of these, but I have no idea how to solve it without any numbers provided. I've tried using $A=\frac{ab}{2} \Rightarrow 2A=ab \Rightarrow 4A^2=a^2b^2$ and incorporating ...
1
vote
2answers
122 views

Prove $(a^2+b^2+c^2)(a+b-c)(b+c-a)(a+c-b)\leq abc(ab+bc+ac)$

Let $a,b,c$ are $3$ edge of a triangle. Prove $(a^2+b^2+c^2)(a+b-c)(b+c-a)(a+c-b)\leq abc(ab+bc+ac)$. My try: I suppose $c=\min\{a,b,c\}$ but I don't know what next.
1
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0answers
34 views

Prove $\frac{3}{64}(ab+bc+ca)^3\geq (de)^3+(ef)^3+(fd)^3$ where $a, b, c$ are three sides of and $d, e, f$ three angle bisectors of a triangle.

A triangle has sides $a, b,c$ and angle bisectors $d, e, f$ where each pair of $a$ and $d$, $b$ and $e$, $c$ and $f$ intersect. Prove that $\frac{3}{64}(ab+bc+ca)^3\geq(de)^3+(ef)^3+(fd)^3$. I was ...
4
votes
2answers
86 views

Finding segment in a right triangle.

Here is the picture of the question: $ABC$ is a right triangle. $m(CBA)=90^\circ$. $m(BAD)=2m(DAC)=2\alpha$. $D$ is a midpoint of $[BC]$. $E$ is a point on $[AD]$. ...
1
vote
1answer
50 views

A problem with “Crossed Ladders Theorem”

In the following diagram, $AY ||BZ$, $AB$ is base. $M$ is $5$ above $N$ and $N$ is $4$ above $O$. What is the height of the triangle $\Delta AOB$. My Work There is a theorem named Crossed Ladder ...
0
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3answers
44 views

Inside an not Equilateral Triangle what is the sum of distances from a random point to 3 sides

Given an not Equilateral Triangle with following side sizes: 45,60,75. Find a sum of distances from a random located point inside a triangle to its three sides. Note 1: Viviani's theorem related only ...
1
vote
1answer
32 views

Inequality in triangle.

If $a,b,c$ are sides of a triangle prove that- $$\frac a{c+a-b}+\frac b{a+b-c}+\frac c{b+c-a}\geq3$$ I am having problem in approaching the problem as the sides are not mentioned as ...
1
vote
3answers
48 views

Circle through the circumcentre of a triangle problem

Let ABC be an acute triangle and O it's circumcentre. Let S denote the circle through A,B, O. The lines CA and CB meet S again at P and Q, respectively. Prove that the lines CO and PQ are ...
0
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1answer
46 views

2011 AMC 12A #13 — Different answers to triangle geometry problem

Triangle ABC has side lengths $AB = 12$, $BC = 24$, and $AC = 18$. The line through the incenter of triangle ABC parallel to $\overline{BC}$ intersects $\overline{AB}$ at M and $\overline{AC}$ at ...
0
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1answer
35 views

New SAT Math Section: Pythagorean Theorem on Soccer Fields

So I attempted this problem and I'm very sure I'm doing it right but I keep getting it wrong as my answer choice is not even one of the answer choices listed. There is a picture that goes with the ...
0
votes
1answer
26 views

Finding a 3rd point in a 3D triangle with known plane, two points and lengths of each side

I have a very similar problem to the below question. right triangle in 3D space, vectors, line intersection? Rather than having the unit vector $A$ I have the lengths $i_2$ to $i_3$ and $i_1$ to ...
0
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0answers
20 views

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 $a \cdot b = \|a\| \|b\| \cos( \theta )$.
0
votes
2answers
461 views

For which $N$ is it possible to alter one side of an equilateral triangle of side length $N$ to get another triangle of integer side lengths, …?

This question is "inspired" by the Rupsa and Equilateral Triangle problem from Code Chef's "October Challenge 2015". The deadline of 12 October 2015 has passed. Given an equilateral triangle having ...
1
vote
0answers
34 views

Joint density of Triangular RV and Maximum of Triangular RVs, parameterised by Uniform RV

Let $x$ be drawn from the uniform distribution on $[0,1]$. $x$ parameterises the Triangular distribution $Y$ with support $[0,1]$. I.e., $$ f_Y(y_i | X = x) = \begin{cases} \frac{2y_i}{x} \quad ...
0
votes
1answer
42 views

moduli space of triangles

I found an article which seems to be aimed for general audience. I couldn't understand sentences about triangles. The link to the article is the following. ...
0
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1answer
354 views

A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
0
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2answers
20 views

Lowest possible value for k for triangle with an integer area

There is a triangle with sides length (9 + k), (39 + k), and (48 + k). The triangle has an area that is an integer. What is the smallest possible value for k? I already tried pythagorean theorem.
2
votes
1answer
94 views

How to find the inverse position inside a triangle [closed]

If I were standing in a triangle - How do I calculate the inverse of my position? Can it be done? It's easy inside a rectangle, but I can't think of how you would do it inside of a triangle. I'm ...