For questions about properties and applications of triangles

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3answers
38 views

A strange contradictive problem

First a part of the set of same balls was arranged into an equilateral triangle, 19 balls were not used, but when the sides of this triangle were needed to be one-uped, it was a 5 ball insufficiency. ...
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2answers
31 views

Is it possible that two triangles satisfy these conditions?

Are there two triangles with equal angles and a pair of equal sides which are not congruent? If yes, please give an example.
2
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1answer
22 views

Prove $a^2\cos B\cos C+b^2\cos C\cos A+c^2\cos A\cos B\leq2S.$

Prove that in any triangle inequality holds: $$a^2\cos B\cos C+b^2\cos C\cos A+c^2\cos A\cos B\leq2S.$$ Is gender inequality that occurs right triangle, not an equilateral triangle. For this reason ...
1
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1answer
17 views

Area of a triangle.

The area of a triangle $ABC$ is $144$.Denote the midpoint of $BC$ by $P$,of $AP$ by $Q$ and of $AC$ by $R$.Calculate the area of the triangle $PQR$. I draw the picture but I do not have any idea to ...
0
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1answer
22 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 ...
5
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1answer
129 views

Symmetrical of a triangle's vertexes

I have the following problem : Show that the symmetrical (ie reflection) of a triangle's vertexes by the opposite side are aligned iff the distance between the orthocenter and the circumcenter is ...
3
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3answers
131 views

Drawing a Right Triangle With Legs Not Parallel to x/y Axes?

I have been presented with an interesting problem. How can I decide whether a right triangle with given side lengths can be placed (with integer coordinate vertices) on a Cartesian plane so that the ...
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0answers
13 views

Properties Of Triangles

If a, b, c be the radii of three circles which touch one another externally, and r1 and r2 be the radii of the two circles that can be drawn to touch these three, prove that 1/r1 - 1/r2 = 2/a + 2/b + ...
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2answers
64 views

Is HHH a congurence criteria for triangles?

I wanted to know if a triangle defined by its 3 heights is unique. I took this up as a challenge but was able to get nowhere, can anyone help me? :)
0
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1answer
30 views

Trigonometric ratios

I'm stuck with a problem. Given is a triangle $\Delta ABC$ with $\angle A = 35°, BC=3$ and $AC=5$. I need to find the two possible values for $\angle C$. I only managed to found one angle. I did the ...
2
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1answer
28 views

When is $3R\le 2h_{\max}$ true for acute triangles?

I was working on a problem recently, and it happened that it could be solved if $3R\le 2h_{\max}$ was true for all acute angled triangles. So I used GeoGebra to check it, and found that for some ...
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1answer
27 views

Find angle and hypotenuse of right angled triangle

Find the missing side and the hypotenuse of a right triangle that has a side length of 5 cm and a perimeter of 30 cm. I'm confused. Can somebody please explain to me how to do this step by step? Not ...
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1answer
36 views

Find the angle of the triangle [closed]

In triangle ABC,median from A bisects BC at D that is BD = CD.Find angle BAD, if angle ADB = 45 degree and angle ACD =30 degree.It is a tough one because it is from RMO (Regional Mathematics ...
3
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2answers
29 views

Triangle similarity question

I've been trying to solve this question for like 40 mins straight and can't seem to get anywhere. I tried drawing a parallel to |KM| from C to |AB| but that didn't seem to help. I just can't see a ...
0
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2answers
20 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 ...
0
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1answer
814 views

Find coordinates of vertex in right triangle

I have a right triangle with known points $A(x_1,y_1), B(x_2,y_2)$ and known cathetus $AC$ and $BA$. I need to find the coordinates of point $C$.
1
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1answer
18 views

Ratio of area between similar triangles

This question has nearly no information and I've been stuck on this for quite some time. I tried drawing the median from A thru G but the 1x to 2x ratio didn't seem to help.
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0answers
24 views

How is the Uniqueness of Equilateral Tetrahedra Proved? [duplicate]

Equilateral tetrahedrons all have this property: For any two of its vertices exists a third vertex, which forms an equilateral triangle with these 2 vertices. (It doesn't necessarily have to be a ...
0
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1answer
29 views

Geometry: Measure of angles

The area of a triangle is equal to 48 cm^2 and two if its sides measure 12 cm and 9 cm, respectively. Find the possible measures of the included angles of the given sides.
2
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0answers
56 views

Beautiful problem about polyhedrons [duplicate]

A regular tetrahedron has this property: For any two of its vertices exists a third vertex, which forms a regular triangle with these 2 vertices. (But it doesn't mean any 3 vertices form a regular ...
1
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0answers
48 views

How to easily prove Euler's theorem, $OI^2=R(R-2r)$?

If $R$ is the circumradius and $r$ is the inradius of some triangle $ABC$, with its circumcenter being $O$ and incenter being $I$, then how to prove: $$OI^2=R(R-2r)$$ I have seen many mentions of ...
0
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1answer
165 views

If an equilateral triangle has an area of 36 units squared, what is the length of a side to the nearest tenth?

I have been working with finding the area of a regular triangles, squares, and hexagons using special right triangle formulas drawn from the radii and apothems, but I cannot for the life of me work ...
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1answer
21 views

Geometry: Solving for the base of the isosceles triangle [closed]

Two altitudes of an isosceles triangle are equal to 20 cm and 30 cm. Determine the possible measures of the base angles of the triangle. Please put a solution. Thank you!
0
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1answer
18 views

Geometry: Finding the sides of the triangle with base and altitude given

The base of an isosceles triangle and the altitude drawn from one of the congruent sides are equal to 18 cm and 15 cm, respectively. Find the lengths of the sides of the triangle. Please help me to ...
0
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2answers
12 views

The vertices of a triangle are A(-1, 1) B(4,0) and C(1,6) Find the equation of the altitude of the triangle ABC drawn from A.

I need some help understanding the process of how you go about answering this question: The vertices of a triangle are A(-1, 1) B(4,0) and C(1,6) Find the equation of the altitude of the triangle ABC ...
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0answers
25 views

How many incongruent isosceles triangles can be found in a grid of certain dimensions? [closed]

How many incongruent isosceles triangles can be found in a grid of dimensions $x\times y$? I thought I had the formula, $$\frac{x-1}{2} (y-1)$$ but this does not work when both $x$ and $y$ are ...
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2answers
37 views

Probability that Three Numbers Drawn Represent Sides of a Triangle

Suppose three numbers are randomly chosen from the following list: \begin{equation} 4,5,7,8,11 \end{equation} What is the probability that the numbers drawn represent sides of a triangle? I posted ...
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0answers
23 views

Triangle length problem using radians [closed]

Triangle PRS is right-angled at R and triangle PQS is isosceles, with PQ = QS. Angle QSR = π/3 radians and RS=3 cm Show that the length of PS is 6√√3+2 cm. (square root of 3 add 2 all square rooted by ...
0
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1answer
23 views

Triangles incident on a vertex (Graphs)

I have a project that I am doing. The specification requires specific methods on a graph class. Two of the methods requires this: 1.numberOfTrianglesIncidentToVertex, calculates and returns the ...
2
votes
3answers
39 views

Trigonometric problem: Elevation angle [closed]

The elevation of the top of a tower $KT$ from a point $A$ is $27^\circ$. At another point $B$, $50$ meters nearer to the foot of the tower where $ABK$ is a straight line, the angle of elevation is ...
0
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1answer
23 views

Demonstrate equality: ON = 2m/m-3 in math exercise

I'm actually getting stuck with a part of a quite tricky math exercise using Thales theorem (I've got difficulties with Thales theorem). In this exercise,you have a right handed Cartesian coordinate ...
0
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1answer
33 views

Point P on side BC of triangle ABC such that PC=2BP. Find ACB if ABC=45º, APC=60º [closed]

Point P on side BC of triangle ABC such that PC=2BP. Find ACB if ABC=45º, APC=60º. I can't solve this one. Tried some stuff but can't work it out. Can this be done using just simple geometry (like ...
0
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1answer
12 views

Is there any polygon exists such that sum of length of square of two adjacent sides is equal to another side/diagonal?

In Right angle triangle we have $ a^2 + b^2 = c^2$ where $a^2 = (x_1-x_2)^2 + (y_1-y_2)^2 ,$ $b^2 = (x_3-x_2)^2 + (y_3-y_2)^2 $and $c^2 = (x_1-x_3)^2 + (y_1-y_3)^2$ And in Square we have $ a^2 + ...
0
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1answer
27 views

Side Lengths of Triangles

Die This is Exercise 3-5 from the Art of Problem Solving Volume 2 by Richard Rusczyk and Sandor Lehoczky. I looked at the solution in the solution manual, but I don't quite understand it, so I'm ...
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2answers
57 views

$ \cos {A} \cos {B} \cos {C} \leq \frac{1}{8} $

In an acute triangle with angles $ A, B $ and $ C $, show that $ \cos {A} \cdot \cos {B} \cdot \cos {C} \leq \dfrac{1}{8} $ I could start a semi-proof by using limits: as $ A \to 0 , \; \cos {A} ...
0
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1answer
16 views

What are the limitations of non-metric distances?

If the triangle inequality does not hold for a distance function (i.e. it is not metric), will this limit its application in some area?
1
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1answer
34 views

$H$ is an orthocenter of triangle $ABC$

$H$ is an orthocenter of angle $ABC$. Angle $B$ is $60^{\circ}$. Perpendicular bisectors of $AH$ and $CH$ cross line $AC$ at points $A_{1}$ and $C_{1}$. Show that the centre of $A_{1}HC_{1}$'s ...
1
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2answers
46 views

Area of one of four regions within a rectangle

There is a figure below (a rectangle). You can see different colors depicting different regions of the figure. The labels on the top of a region defines the area of that region. Can you find the ...
3
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1answer
61 views

Ortocenter and incenter

In triangle $ABC$: $H_{1}$ is a foot of an altitude from side $BC$, $H_{2}$ is a foot of an altitude from side $AC$, $H_{3}$ is a foot of an altitude from side $AB$, $M_{1}$ is midpoint of $BC$, ...
0
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1answer
59 views

Disprove the possibility of such a triangle.

The image is not that good, but, consider the following figure to be true without actually constructing it,how can one person find a $fault$ in it. The blue colour represents perpendicular, The ...
0
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0answers
13 views

Determine Angle based on Vertical Displacement

I need a formula that will help me identify a observing angle based on the following example: Launching a bottle rocket. Test 1: looking from 35 degree angle, when a bottle is launched from ground, ...
1
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1answer
24 views

Triangle Inequality Property for the Euclidean Metric

I've read in many of my books that the triangle inequality for a metric space of the Euclidean Metric is defined as: $$d(x,y) \leq d(x,z) + d(z,y)$$ But when I look up the proof, to help me ...
15
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4answers
164 views

A Triangle Determinant

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
0
votes
2answers
42 views

double integral over an arbitrary triangle

Assume we have an arbitrary triangle ABC in x-y plane and we want to integrate a function $f(x,y)$ over surface of this triangle as shown in fig. 1: We can define another coordination system [x' ...
0
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2answers
28 views

Diophantine Equation Related To Triangles

a,b and c are the sides of a triangle and a, b, c are integers. I need to solve the following Diophantine equation for positive integral values of k. $bc(b+c-a) = k^{2}(a+b+c)$ I think some ...
-1
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2answers
38 views

In every triangle the equation $\overrightarrow{TA}+\overrightarrow{TB}+\overrightarrow{TC}=\overrightarrow {0}$ holds [closed]

If $T$ is the point where the expected medial triangle i.e. $\overrightarrow {AA_1}\cap\overrightarrow {BB_1}\cap\overrightarrow {CC_1}=\{T\}$, prove that the equation ...
0
votes
1answer
41 views

Dividing a triangle into seventeen equal parts.

I was trying to solve a problem on Pigeonhole principle from Problem Solving Strategies by Arthur Engel. A target has the form of an equilateral triangle with side 2 units. If it is hit ...
0
votes
1answer
22 views

Trigonometry: Find the side of a triangle within a triangle

Please help. I found a solution to this problem on yahoo answers but I do not understand the answer. I would use the laws of cosine but I have to be able to answer this without a calculator If AB = ...
1
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2answers
73 views

What are the ranges of triangle angles?

Lets say, that $\alpha \le \beta \le \gamma$. As shown here, $60 \le \gamma \lt 180$. What are the minimum and maximum values of $\alpha$ and $\beta$? The answer: $$0\lt \alpha \le 60 \\ 0 \lt ...
0
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1answer
26 views

How to prove two triangles have the same centroid?

Suppose you have a triangle ABC and three similar exterior triangles BCX, CAY and ABZ. How can I prove that the centroids of ABC and XYZ are the same point?