For questions about properties and applications of triangles

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Inequality of area of two triangles

Let $ABC$ be a triangle with sides $a,b,c$ and $A_1B_1C_1$ be another triangle with sides $a+\frac{b}2$, $b+\frac{c}2$, $c+\frac{a}2$. Prove that: $$\frac94[ABC]\le[A_1B_1C_1]$$ I tried using ...
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3answers
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Maximum value of $\sin A+\sin B+\sin C$?

What is the maximum value of $\sin A+\sin B+\sin C$ in a triangle $ABC$. My book says its $3\sqrt3/2$ but I have no idea how to prove it. Can anyone help? :)
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Proving a triangle equilateral given condition $al_a^2+bl_b^2+cl_c^2=9R\Delta$

$ABC$ is a triangle, with $l_a$, $l_b$, $l_c$ as angle bisectors, $R$ as circumradius and $\Delta$ as area, such that: $$al_a^2+bl_b^2+cl_c^2=9R\Delta$$ Is it true that $ABC$ is equilateral? I am ...
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1answer
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To calculate side of the Equilateral triangle

The figure is an equilateral triangle. 3 line segments , which meet at a(any) point in the triangle , are of the length 5cm, 4cm, and 3 cm as shown in the figure. Find the side of the equilateral ...
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Prove that OD is a the angle bisector of the angle BOC.

Let ABC be a non-isosceles triangle and I be the intersection of the three internal angle bisectors. Let D be a point of BC such that $ID \perp BC$ and O be a point on AD such that $IO \perp AD$. ...
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Given a Triangle ABC, draw squares ACDQ and BCER outside on the two sides of the triangle. Prove that.. [on hold]

Start with Triangle ABC, draw squares ACDQ and BCER outside on the two sides of the triangle. Prove DE is 2x length of median corresponding to vertex C.
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Right Triangles

Right triangle ABC has hypotenuse AC, angle CAB=30°, and BC=√2. Right triangle ACD has hypotenuse AD and angle DAC=45°. The interiors of ABC and ACD do not overlap. Find the length of the ...
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Triangle Segments

In a triangle $ABC$, $AB=5$, $BC=16$, $AC= \sqrt{153}$, and $D$ is on segment $BC$. Compute the sum of all possible integral measures of $AD$. I've been having trouble trying to solve this problem ...
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find median length when knowing the side length and its angle

I have isosceles triangle. Its equal sides size are known. the angles that the sides make with a base are known. What is the equation to find out the median length that is perpendicular on the ...
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Beautiful little geometry problem about sines

Given triangles ABC and $A_1B_1C_1$ such that $\sin A = \cos A_1, \sin B = \cos B_1, \sin C = \cos C_1$. What are the possible values for the biggest of these 6 angles? I tried some stuff like sine ...
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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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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.
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1answer
23 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 ...
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1answer
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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 ...
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1answer
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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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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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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? :)
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1answer
31 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
30 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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3answers
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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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1answer
28 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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2answers
22 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 ...
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1answer
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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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2answers
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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 ...
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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 ...
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1answer
37 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 ...
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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 ...
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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 ...
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1answer
35 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.
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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 ...
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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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2answers
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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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1answer
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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 ...
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1answer
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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 ...
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1answer
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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 ...
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1answer
130 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 ...
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1answer
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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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1answer
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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 + ...
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1answer
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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?
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$H$ 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 ...
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2answers
47 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 ...
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1answer
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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 $BC$, ...
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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, ...
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1answer
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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 ...
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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 ...
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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' ...
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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} ...
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1answer
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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 ...
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1answer
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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 = ...
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1answer
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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?