For questions about properties and applications of triangles

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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 ...
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1answer
33 views

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

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 ...
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1answer
356 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 ...
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1answer
32 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 ...
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1answer
55 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 ...
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0answers
42 views

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

In a triangle with sides $a, b,c$ and the relation $a^3=b^3+c^3$, get the angle between $b$ and $c$.
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2answers
39 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 ...
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3answers
31 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.
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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 ...
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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 ...
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0answers
39 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?
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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?
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3answers
46 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 ...
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2answers
35 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 ...
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2answers
39 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 ...
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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 ...
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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 ...
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2answers
119 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.
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0answers
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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 ...
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2answers
84 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 $[AD]$. ...
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1answer
47 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 ...
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1answer
825 views

How to know which side of the right angled triangle is the base? [closed]

If we are given a right angled triangle without any angle or length of any side. How we will find that which side is the base, which side is the perpendicular.
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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 ...
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1answer
31 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 ...
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3answers
47 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 ...
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1answer
44 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 ...
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1answer
34 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 ...
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1answer
25 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 ...
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0answers
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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 $a \cdot b = \|a\| \|b\| \cos( \theta )$.
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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 ...
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0answers
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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 ...
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1answer
40 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. ...
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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 ...
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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.
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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 ...
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7answers
47k views

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 ...
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4answers
186 views

Finding area problem

There is this simple geometry question that seems so easy but I think the question lacks some information (does it?). Or maybe there are other ways to solve the problem. So the problem says, there ...
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2answers
39 views

Finding an angle between two vectors

I am trying to answer part $d)$ by using my answer to part $c)$. From what I can see, the only possible way to do this is to find the lenght of $AB$ and $OB$, and, using the angle in part $c)$, apply ...
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1answer
57 views

Triangle with same black and white areas

Suppose we have an infinite chessboard with the usual black/white coloring. A triangle $T$ with area $a$ is given with vertices at corners of some cells. Prove that there exists another triangle $T'$ ...
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1answer
23 views

$3$ Triangles and a quadrilateral

In the following diagram, in $\Delta ABC$, $CD$ and $BE$ are two cevians intersecting it point $O$. Area of $\Delta BOD = 3, \Delta BOC = \Delta COE = 7$. What is the area of $ADOE$. Note: I can't ...
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3answers
58 views

How to determine (and explain) the sum of angles without measuring?

Below is a photo of the angles/triangles in which I am working on determining the sum of the angles without measuring. The angles are a,b,c,d,e,f. I understand that angles are formed my ...
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3answers
48 views

Find the two other sides in a 15-30-135 triangle

A triangle has angle measures of 15, 30, and 135 degrees. The side opposite the 15 angle is x feet, the side opposite the 30 angle is y feet, and the side opposite the 135 angle is 2 feet. Find x and ...
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2answers
41 views

Prove $||a| - |b||$ is less than or equal to $|a-b|$

I was given the hint to split it into two cases ($|a| - |b|$ being positive and negative) and then use the triangle inequality. However, since the triangle inequality says that $|a+b|$ is less than or ...
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2answers
364 views

Prove triangles formed by two midpoints and an altitude are congruent

Triangle ABC has altitude BH. M is the midpoint of AB, and N is the midpoint of CB. Prove triangle MBN is congruent to triangle MHN. Can we say that MN bisects BH? If so, why? If MN bisects BH (at ...
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4answers
95 views

Prove that triangle is equilateral

I have a triangle here, how do I prove that $BCD$ is equilateral(so all lines have the same length) And yes this is 2D What I have so far is $$BAC = 120^\circ$$ So how do I point out that $$BCD ...
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3answers
49 views

Prove the triangle is equilateral

HINTS ONLY please. This is very confusing right off the bat. My guess was that we show the angle $C, M, N$ are all $60^{\text{o}}.$ But I am having difficulty doing as as none of the givens have ...
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3answers
2k views

“World's Hardest Easy Geometry Problem”

This question is a "corollary" (if you will) to the World's Hardest Easy Geometry Problem (external website). Formally, this is called Langley's Problem. The objective of that problem was to solve for ...
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0answers
42 views

How to find a triangle's perimeter only using base and height?

Without measuring the length of the other two sides, is there a way to find the perimeter with one side (Base) and the height of that side?
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1answer
414 views

How to find heading angle to an object who's 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 ...
2
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1answer
86 views

At what angle does the stone need to be hit?

In curling, it is often necessary to hit and displace an opponent’s stone to win the end. Olivia would like to hit her opponent’s stone with her own stone. If she releases her stone at the hog line, ...