For questions about properties and applications of triangles

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1answer
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Question on circles…

If three circles with radii ${3}$,${4}$,${5}$ touch each other externally at points P,Q and R,then the CIRCUMRADIUS of ∆PQR is...?? My attempt i think that the let the point of the common ...
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0answers
28 views

How can I find the distance between two points within a triangle if I have the distance between each point and each vertex of the triangle?

Title says it all. It would be useful to extend the question to finding the distance if any of the points is outside of the triangle, but I'm trying to figure out the basic problem first.
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1answer
34 views

“product” of triangles

I consider 2 triangles $T_1$ and $T_2$ in a plane (as surfaces and not circumferences) What is the geometrical shape of the set S of points P=(w,z) such that there exists 2 points : $P_1=(x_1,y_1) \in ...
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0answers
32 views

How to determine if a triangle is inside another triangle without any intersecting sides

This question is for getting the right logic down for a programming task. I need to be able to determine if a triangle is located inside another without any sides intersecting each other. The two ...
0
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1answer
35 views

Distance from Chicago to New York

An airplane flies $520$ miles from Chicago to Virginia. Then it turns $45$ degrees to face New York and flies $630$ miles to New York. What is the distance from Chicago to New York? Given the $45$ ...
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Integer Triangle Radicals conjecture

An integer sided triangle has an area $A$. Heronian triangle areas have no radical, or radical 1. Otherwise, $4 A$ will always be of the form $a\sqrt{r}$, where $r$ is the squarefree radical of the ...
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1answer
38 views

Is proof of Pythagoras Theorem using Similarity circular?

Please see this link. (hope it doesn't rot) Is this proof circular? I think similarity is proved by basic laws of trigonometry, especially Pythagoras theorem. When I search for proof of Similarity, ...
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1answer
33 views

How to setup vector story problems

I'm studying for my trig final and I know how to do all the math, but I don't always understand how to setup the story problems. Mostly I'm struggling with vector story problems. For example: Forces ...
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2answers
45 views

In equilateral triangle,One vertex of a square is at the midpoint of the side, and the two adjacent vertices are on the other two sides of triangle

In the equilateral triangle $ABC,AB=12.$One vertex of a square is at the midpoint of the side $BC$, and the two adjacent vertices are on the other two sides of the triangle.Find the length of the side ...
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1answer
40 views

Parallelogram -diagonal-similarity problem

Given a parallelogram $ABCD$ . Points $M$ and $N$ are respectively the midpoints of $BC$ and $CD$ . Lengths $AM$ and $AN$ intersecting diagonal $BD$ consecutive points $P$ and $Q$. Prove that ...
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1answer
18 views

similar triangle problem in parallelogram with vertical lines

Can anyone help me with this task? I have no idea how to start. From the top $B$ of a parallelogram $ABCD$ lowered the vertical $BP$ and $BQ$ on the directions of $AD$ and $CD$ . From the top $D$ ...
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1answer
28 views

Let $K$ be midpoint of the hypotenuse of a right triangle $ABC$.On the leg $AB$ is a point $M$ s.t $BM=2MC$.Show that $MAB$ and $MKC$ are similar.

Let $K$ be the midpoint of the hypotenuse of a right triangle $\triangle ABC$. On the leg $BC$ is a point $M$ such that $ BM = 2MC$ . Prove that the triangles $\triangle MAB$ and $\triangle MKC$ ...
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0answers
38 views

If the circumcircle of a triangle cuts its nine point circle orthogonally,then prove that $\cos A\cos B\cos C=\frac{-1}{2}$

If the circumcircle of a triangle cuts its nine point circle orthogonally,then prove that $\cos A\cos B\cos C=\frac{-1}{2}$ I know that two intersecting circles are orthogonal if any one of the ...
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2answers
57 views

Prove $\sin(A-B)/\sin(A+B)=(a^2-b^2)/c^2$ [closed]

prove For any triangle $\triangle ABC$, prove that $$\frac{\sin(A-B)}{\sin(A+B)}=\frac{a^2-b^2}{c^2}$$
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1answer
62 views

Draw A Triangle From 3 Excenters and Ex-radii

My teacher gave me this problem and told me to think- " Is it possible to draw a triangle, given the three ex-centers and length of the ex-radii?" I don't know if it's possible or not. So, my ...
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2answers
47 views

Geometry experts! Three equal tangential circles: What is the ratio of the blue line to the red line?

Consider the three tangential circles of equal radii inscribed in the equliateral triangle (linked to below). What is the ratio of the blue line to the red line? The red line is simply the diameter ...
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0answers
58 views

Prediction Interval from Markov Chains

Thank you for taking the time to look at my question. Short, less involved question: How do you find Prediction Intervals with non-Gaussian residuals? Here is the situation: I have made a model that ...
0
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1answer
15 views

Find the triangle with the greatest area using trigonometric ratios

The hypotenuse, c, of right $\triangle$ABC is $7.0$cm long. A trigonometric ratio for angle $A$ is given for four different triangles. Which of these triangles has the greatest area? a) sec $A$ = $1....
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1answer
33 views

Linear algebra - Proof of a thesis concerning the height in triangles!

My Math teacher gave us some tasks we should work on. I solved most of them already, however I still could not manage to figure out the solution for this one! I would really appreciate, if someone of ...
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1answer
36 views

Prove that $\sin^2\frac{A}{2}\csc2A$, $\sin^2\frac{B}{2}\csc2B$, $\sin^2\frac{C}{2}\csc2C$ are in harmonic progression

If sides $a,b,c$ of $\triangle ABC$ are in arithmetic progression (AP), then prove that $$\sin^2\frac{A}{2}\csc2A, \quad\sin^2\frac{B}{2}\csc2B, \quad \sin^2\frac{C}{2}\csc2C$$ are in harmonic ...
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1answer
42 views

Prove that there are two values to the third side,one of which is $m$ times the other.

Let $1<m<3$. In $\triangle ABC$, if $2b=(m+1)a$ and $\cos A=\frac{1}{2}\sqrt{\frac{(m-1)(m+3)}{m}}$, prove that there are two values to the third side, one of which is $m$ times the other. $\...
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2answers
43 views

Find $\frac{\cos A}{p_1}+\frac{\cos B}{p_2}+\frac{\cos C}{p_3}=$

Let $p_1,p_2,p_3$ be the altitudes of $\triangle ABC$ from vertices $A,B,C$ respectively, $\Delta$ is the area of the triangle,$R$ is the circumradius of the triangle,then$\frac{\cos A}{p_1}+\frac{\...
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1answer
28 views

What is the length of the shorter trisector of the right angle in a $3$-$4$-$5$ triangle?

What is the length of the shorter trisector of the right angle in a $3$-$4$-$5$ triangle? I found this question in a local question paper, and I am unable to solve it. I applied Cosine formula, but ...
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1answer
31 views

Two triangles in a plane

Let $\Delta_1$ and $\Delta_2$ be two triangles in a plane with centroids $G_1$ and $G_2$ respectively. Let $X$, $Y$ be variable points on the perimeter of the triangles $\Delta_1$,$\Delta_2$ ...
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1answer
37 views

Prove that the center of a circle within a constructed triangles lies on the angle bisector

I was given steps to construct a figure: 1.) Construct a horizontal ray AB and a segment AC at an angle to the ray. Locate point D anywhere on ray AB and construct the segment CD. 2.) Construct the ...
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2answers
73 views

If three cevians are concurrent at a point and form triangles of equal area, the point is the centroid

Let D,E,F be points on side BC,CA,AB of triangle ABC. The three cevians are concurrent at a point G. The areas of triangles BGD, CGE and AGF are equal. Prove that G is the centroid of ABC I have ...
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1answer
35 views

Vector addition and Pythagorean theorem

Finding length or magnitude using vector addition and the Pythagorean theorem. I am trying to understand why vector addition and the Pythagorean theorem are giving different results? Vector ...
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0answers
17 views

Relative velocity- Finding the direction of wind.

An aircraft is flying due south at $350~\text{kmh}^{-1}$. The wind is blowing at $70~\text{kmh}^{-1}$ from the direction of $\theta$, where $\theta$ is acute. Given that the pilot is steering the ...
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1answer
36 views

Changing the side of a triangle without changing area?

$\triangle ABC$ has vertices $A=(8,2)$, $B=(0,6)$ and $C=(-3,2)$. Point $C$ can be moved along a certain line with points $A$ and $B$ remaining stationary so that the area of $ABC$ will not change? ...
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0answers
14 views

How to find the third point of a triangle in a 3D space (arm rig)

I am attempting to create a system that will replicate arm movement, so far I have mastered this in a 2D plane however I am having trouble adding the third dimension. Here is what is given, You know ...
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1answer
30 views

How do I calculate the third point of a triangle in a 3Dimensional Plane

I am attempting to create a system that will replicate arm movement, so far I have mastered this in a 2D plane however I am having trouble adding the third dimension. Here is what is given, You know ...
0
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1answer
17 views

How to calculate the X distance while Z object is moving top to bottom?

First of all, sorry for my extremely low knowledge about mathematics. All i am able to do is this image which describe the problem. Image of triangle + information I want to know that how can i ...
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1answer
21 views

External Bisectors of Triangle ABC

The exterior angle bisectors of $\angle B$ and $\angle C$ intersect on point $O$. $\angle BOC=70°$. Find $\angle OAC$.
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0answers
40 views

How many different ways can a circle intersect a triangle N ways?

Consider a circle intersecting a triangle. The circle and triangle can have between 0-6 total intersection points. Is there a mathematical formula for the number of possible ways they can intersect ...
1
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1answer
29 views

Proof to show that a quadrilateral is a paralellogram.

I am new to this site and I don't know how to code yet, so bear with me. I am trying to show that the quadrilateral ABCD in the picture below is a parallelogram. I know that since triangle EBC is ...
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2answers
36 views

Which sides of a triangle are visible to an observator?

Working o 2-d plane. Supposing that there is a observer standing on the origin (0, 0) looking to the first quadrant. If there is a triangle drawn on the first quadrant, what sides are visible to the ...
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1answer
27 views

In a triangle $ABC$ with $A=(1,3) ,B =(q,0), C =(p,-4)$ [closed]

Let $A=(1,3),B =(q,0), C =(p,-4)$, with $p>0$, the slope of $AB$ is $+45^\circ$ and $AC= \sqrt{50}$. Determine the gradient of $AB$ Calculate the equation of the line $AB$ Calculate the value of ...
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3answers
168 views

Similar triangle proof in parallelogram

Can anyone help me with this task. From the top of a parallelogram $ABCD$ lowered the vertical $AM$ and $AN$ on the lines BC and CD . Prove that triangles $\triangle ABC$ and $\triangle AMN$ similar ....
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2answers
27 views

Prove that DE || BC

Let M be the midpoint of side BC in triangle ABC. The angle bisector of BMA intersects AB in D, while the angle bisector of CMA intersects AC in E. How can i prove that DE||BC? I drew out the ...
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2answers
43 views

Is there a way of determinine the side lengths of a isosceles triangle knowing its angles and area?

I want to be able to determine the side lengths (or at least one side length) of an isosceles triangle knowing only its surface area and angles. Is this possible?
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1answer
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How many different shapes can you construct with n equilateral triangles?

If you have n equilateral triangles, and you want to connect them all to each other at the edges, how many different shapes can you make? Triangles are identical in size and shapes that are ...
2
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5answers
69 views

what is the value of angle A

The triangle ABC is random. The line $AD$ is twice big as the line $DC$ ($AD=2*DC$). We know only the two angles that are shown in the picture. What's the value of angle $A$?
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1answer
31 views

Median BM of triangle ABC two results

Given Calculate the measure of the median $\overline{BM}$ of ABC triangle, given A (-6.1); B (-5,7) and C (2,5) I get this result: $Xm = \frac{Xc - Xa}{2} + Xa$ $Xm = \frac{2-(-6)}{2} + (-6) = ...
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11answers
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In a right triangle, can $a+b=c?$

I understand that due to the Pythagorean Theorem, $a^2+b^2=c^2$, given that $a$ and $b$ are legs of a right triangle and $c$ is the hypotenuse of the same right triangle. However, most of the time, $a+...
2
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0answers
57 views

Prove that the intersection point of lines $AK$ and $CL$ lies on the line $BO$

$AA', BB'$ and $CC'$ heights of an acute triangle $ABC$. The circle with center $B$ and radius $BB'$ intersects the line $A'C'$ in the points $K$ and $L$. Prove that the intersection point of lines $...
2
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1answer
22 views

All triangles that have the same orthocenter and circumcircle have the same nine-point circle

True or false? Prove it. I guess it would help to figure out whether 2 triangles can have the same circumcenter or orthocenter and not be congruent. I have no clue how to figure this out. If they ...
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0answers
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A new family circle associated with the Tucker hexagon and the Symmedian point

I am looking for the problem following: Let ABC be a triangle, let $A_1B_1C_1$ be a cevian triangle of the symmedian point. Let $B_aC_aC_bA_bA_cB_c$ be a Tucler hexagon of $ABC$. Such that $A_bA_c ...
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0answers
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A generalization of the first Droz-Frany circle

I am looking for a proof of the following problem: Let $ABC$ be a triangle with circumcenter $O$, and the medial triangle $M_aM_bM_c$. Let $O_a, O_b, O_c$ be three points on three lines $OA, OB, ...
2
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1answer
39 views

What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles?

What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles? My work so far: Case 1) let $ABC -$ an acute-angled triangle. Then radii of the ...
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1answer
38 views

How to solve this problem about the reciprocals of the angles inside an acute triangle?

I was solving some previous year papers and stumbled upon this question. I don't even know how to approach this problem. The problem is as follows: Consider an acute angled triangle $PQR$ such ...