For questions about properties and applications of triangles

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2
votes
2answers
38 views

How to parameterize the interior of a triangle

I would like to know how to parameterize a triangle over $[0,1] \times [0,1]$. I actually only care that the mapping is surjective but a bijection is always nice I suppose. I found this in which an ...
4
votes
0answers
67 views

Similar Triangle dissections

Andrzej Zak found that a triangle with sides based on powers of the root $d^6-d^2-1=0$, $(d=1.15096...)$ that can replicate itself with 6 differently sized copies. The numbers are powers of $d$. The ...
-1
votes
0answers
11 views

Get location / position of an object via 2 cameras

Best In the following image, you can view the setup of my problem. In general, i've 2 cameras which have a view-angle of 48° and 64°. Secondly, I know the position of my camera's (which means i can ...
-1
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1answer
18 views

Question on proving congruent triangles [on hold]

How can I prove triangle ABD congruent triangle ACE? I guess the reason should be SAS, but I dont know how to prove the angles the same. ABC and ADE are equilateral triangles.
6
votes
6answers
3k views

Why is not possible to draw this triangle?

Why is it not possible to draw triangle $DEF$ with $EF=5.5cm$,$\angle E=75^0$ and $DE-DF=1.5cm$?(I used this method for ...
-1
votes
2answers
52 views

Triangle angle question [on hold]

I need help about a triangle angle question.
-2
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3answers
160 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 ...
3
votes
1answer
249 views

History of incenter and Euler line

It is easy to see that if a triangle is isosceles, then its incenter lies on its Euler line. Who first proved the converse of this result and what technique was used? (See the post "The incenter and ...
0
votes
1answer
21 views

For any point inside a triangle

Let $P$ be an interior point of the triangle $\triangle ABC$. Assume that $AP$, $BP$ and $CP$ meet the opposite sides $BC$, $CA$ and $AB$ at $D$, $E$ and $F$, respectively. Show that $\frac{AF}{FB} ...
1
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0answers
33 views

An interesting geometry problem with angle bisectors and tangent

I have found the following problem: There is an acute $\triangle ABC$. Denote its circumcircle as $\omega$. The angle bisector of $\angle BAC$ intersects $BC$ and $\omega$ in points respectively $A_1$ ...
2
votes
1answer
29 views

Distance between incentre and orthocentre.

I want to prove that the distance between incentre and orthocentre is $$\sqrt{2r^2-4R^2\cos A\cos B\cos C} $$here $r$ is inradius and $R$ is circumradius. I considered $\triangle API$ ($P$ is ...
8
votes
2answers
1k views

How to derive the law of cosines without the pythagorean theorem

To me, it seems that the Pythagorean theorem is a special case of the law of cosines. However, all derivations that I can find seem to use the Pythagorean theorem in the derivation. Are there any ...
5
votes
2answers
266 views

How do squares of non-right triangles relate?

How do squares of the sides of a triangle, any triangle, relate?
2
votes
3answers
10k views

Calculate coordinates of 3rd point (vertex) of a scalene triangle if angles and sides are known.

I am writing a program and I need to calculate the 3rd point of a triangle if the other two points, all sides and angles are known. ...
0
votes
2answers
43 views

2 circles in an isosceles triangle

I've been given the following school problem: ABC is an isosceles triangle (AB = AC). The radius of the incircle is R and of the other circle (which is tangent to the incircle and to the legs of ...
1
vote
2answers
33 views

Triangle group $(\beta,\beta,\gamma)$ is a subgroup of the triangle group $(2,\beta, 2\gamma)$.

Let $1/\alpha+1/\beta+1/\gamma<1$, and let us consider the triangle group $(\alpha,\beta,\gamma)$, i.e. the subgroup of $\mathbb{P}\mathrm{SL}(2,\mathbb{R})$ induced by the hyperbolic triangle ...
-1
votes
1answer
24 views

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 ...
11
votes
0answers
1k views

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 ...
-2
votes
1answer
32 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 ...
1
vote
0answers
17 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.
0
votes
0answers
27 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 ...
1
vote
2answers
412 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 ...
0
votes
1answer
30 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 $45$ ...
0
votes
2answers
35 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 ...
0
votes
1answer
897 views

Calculate 3rd point of a triangle, given 2 points and all angles in 2D

I have stumbled upon an interesting problem. I tried to find an answer here but there are just too many similar threads which did not really help me, so I was trying to figure it out by myself. The ...
1
vote
0answers
24 views

Is proof of Pythagoras Theorem using Similarity circular?

Please see this link: https://www.math.nmsu.edu/~breakingaway/Lessons/PTUST/PTUST.html (hope it dosent rot) Is this proof circular? I think similarity is proved by basic laws of trigonometry, ...
0
votes
1answer
25 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 ...
0
votes
1answer
435 views

What is the maximum number of triangles in a planar graph with n vertices?

The answer is obvious for small numbers of nodes: $$n<3: 0\\ n=3: 1\\ n=4: 3\\ n=5: 5 (see below)$$
176
votes
8answers
22k views

V.I. Arnold says Russian students can't solve this problem, but American students can — why?

In a book of word problems by V.I Arnold, the following appears: The hypotenuse of a right-angled triangle (in a standard American examination) is 10 inches, the altitude dropped onto it ...
1
vote
1answer
31 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 ...
1
vote
1answer
14 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 $D$ ...
0
votes
1answer
22 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 $\triangle MKC$ ...
0
votes
2answers
104 views

Geometric proof with a isosceles triangle

Given is $\triangle ABC$ with the medians $AD$, $BE$ with $|AD|=|BE|$. The medians intersect in $S$. a. Use similar triangles to show that $|AS|:|SD|=|BS|:|SE|=2:1$. b. Prove that $\triangle ABC$ is ...
1
vote
1answer
60 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 ...
0
votes
0answers
32 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 ...
-3
votes
2answers
54 views

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

prove For any triangle $\triangle ABC$, prove that $$\frac{\sin(A-B)}{\sin(A+B)}=\frac{a^2-b^2}{c^2}$$
0
votes
2answers
39 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 ...
1
vote
0answers
56 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 ...
19
votes
11answers
5k views

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, ...
0
votes
1answer
14 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 $A$ = ...
0
votes
1answer
30 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 ...
1
vote
1answer
34 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 ...
1
vote
1answer
32 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. ...
1
vote
2answers
138 views

Configuration of five or more mutually equidistant points in space.

How is it proved that there is no configuration of five or more mutually equidistant points in $R^3$? Is it done by induction? I'm stuck. Help would be appreciated. Well, surely equilateral ...
-3
votes
3answers
46 views

Prove the Sine Rule [closed]

Suppose that $ABC$ is a triangle. Prove the sine rule: $d(A, B)/\sin(∠ACB) = d(B, C)/\sin(∠BAC) = d(C, A)/\sin(∠CBA)$. I am unsure really how to proceed with this.
0
votes
2answers
41 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 ...
3
votes
2answers
436 views

Barycentric coordinates of a triangle

I have to do what described in the picture below. Consider the planar triangle $[p_1,p_2,p_3]$ with vertices $p_1=\begin{pmatrix}-2\\-1\end{pmatrix}$, ...
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0answers
41 views
0
votes
1answer
26 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, ...
0
votes
1answer
29 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 $\Delta_1$,$\Delta_2$ ...