For questions about properties and applications of triangles

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4answers
398 views

In the triangle ABC, D and E are points of trisection of segment AB; F is the midpoint of segment AC. What is the ratio: MN/BF

This is a euclidean geometry problem. No angles measures are given. There are no right angles given. DE/AB = 1/3; AF = FC. I have tried countless extensions and constructions betond what is shown to ...
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2answers
52 views

Usage of law of sines

The vertex angle of an isosceles triangle is 35 degrees. The length of the base is 10 centimeters. How many centimeters are in the perimeter? I understand the problem as there are two sides with ...
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1answer
69 views

Trigonometry (non right angled triangles)

The height of a vertical tower is to be found by a surveyor. The angle of elevation of the top of the tower from a point on the horizontal ground some distance away is measured as 28.7 degrees. From ...
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2answers
4k views

Can an equilateral triangle be an isosceles triangle, too?

I've looked in a math book that an isosceles triangle has at least two congruent sides. I also know that the words "at least" mean this symbol: $\ge$, which means "is greater than or equal to" or "is ...
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2answers
144 views

Beautiful triangle problem

Circle, inscribed in $ABC$, touches $BC, CA, AB$ in points $A', B', C'$. $AA' BB', CC'$ intersect at $G$. Circumcircle of $GA'B'$ crosses the second time lines $AC$ and $BC$ at $C_A$ and $C_B$. Points ...
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3answers
6k views

How is the hypotenuse the longest side of any right triangle?

I see that the hypotenuse of a right triangle is opposite the right angle, but how is it always the longest side? I also know that it connects to endpoints of other sides. Please help me out with ...
3
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1answer
49 views

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
157 views

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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2answers
79 views

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
56 views

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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2answers
116 views

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 ...
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1answer
43 views

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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2answers
44 views

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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2answers
73 views

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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3answers
57 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
58 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
28 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
42 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 ...
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1answer
433 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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2answers
86 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? :)
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1answer
37 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 ...
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1answer
37 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
501 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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2answers
163 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
142 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
49 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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2answers
142 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 ...
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0answers
68 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 ...
2
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0answers
235 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 ...
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1answer
292 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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1answer
3k 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 ...
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2answers
40 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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2answers
208 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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1answer
144 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 ...
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1answer
42 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 ...
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1answer
51 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 ...
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1answer
144 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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3answers
111 views

Complex numbers on circle of unit radius

Given three points in the complex plane (i.e. numbers $z_1,z_2,z_3\in\mathbb C$), they define a unique circle (unless they are collinear). When does that circle have radius one? I know how to compute ...
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1answer
56 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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1answer
32 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 + ...
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1answer
46 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?
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1answer
63 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 ...
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2answers
389 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
84 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 $BC$, ...
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1answer
103 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 ...
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2answers
56 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 ...
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2answers
380 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' ...
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2answers
77 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
372 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 ...
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1answer
237 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 = ...