For questions about properties and applications of triangles

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4
votes
4answers
360 views

Triangle-free graph with 5 vertices

What is the maximum number of edges in a triangle-free graph on 5 vertices? No answers, please...just hints. I believe that E $\leq$ 5, but I'm not sure where to go from there.
5
votes
3answers
172 views

Angle in a triangle within a circle.

A and B are two points on the circumference of a circle with centre O. C is a point on OB such that AC $\perp OB$. AC = 12 cm. BC = 5 cm. Calculate the size of $\angle AOB$, marked $\theta$ on the ...
2
votes
0answers
40 views

Lemoine Point triangle

from Wolfram MathWorld, I know there is a Lemoine point of triangle, also called symmedian point, the sum of squared distances of this point to all the three sides is algebraically minimum. How can I ...
0
votes
4answers
81 views

In triangle ABC, Find $\tan(A)$.

In triangle ABC, if $(b+c)^2=a^2+16\triangle$, then find $\tan(A)$ . Where $\triangle$ is the area and a, b , c are the sides of the triangle. $\implies b^2+c^2-a^2=16\triangle-2bc$ In triangle ...
2
votes
0answers
49 views

Intesection point of feet of altitudes

If triangle has vertexes at $(x_1,y_1),(x_2,y_2),(x_3,y_3)$, is the intersection points of feet of altitudes $$x_h = \frac{x_1x_2(y_2-y_1) + x_2x_3(y_3-y_2) + x_3x_1(y_1-y_3) + y_1^2(y_3-y_2) + y_2^2(...
0
votes
1answer
30 views

Finding the minimum value of squares of sides of a quadrilateral

What is the minimum value of $\frac{a^2+b^2+c^2}{d^2}$ where $a,b,c,d$ are the sides of quadrilateral I assumed the diagonals to be $p$ and $q$. I got that for minimum angle $A$ and $C$ must be $90^{\...
0
votes
2answers
56 views

Length of a right triangle created by skewing a rectangle's edge by a fixed amount

I have the above problem for a grid-based graphics system I'm working on, and I'm not sure if the math is solvable or not. I'm trying to determine the value of $A$. I've attempted to use ...
0
votes
1answer
66 views

Finding the value of k

If $x,y,z$ are perpendicular distances from circumcenter on the sides $BC,AC$ and $AB$ respectively. In need find $k$ such that $$\frac ax+\frac by+\frac cz=\frac{abc}{kxyz}$$ (Lowercase letters ...
1
vote
1answer
107 views

Finding the third side of a triangle, given ratio of two sides and difference of two angles [closed]

Given $a=2b$ and $|\angle A-\angle B|=60$ degrees. Find the third side, where lowercase letters denote opposite sides and uppercase letter angles. Progress I could find the $\cos C$ but then got ...
0
votes
2answers
68 views

Prove that $\frac{1}{2}ab \equiv \int_0^b \! f(x) \, \mathrm{d}x$ when calculating the area of a right triangle.

Triangle $ABC$ is a right triangle with sides $AB$, $BC$ and $AC$. $a$ is the length of $AB$. $b$ is the length of $BC$. $c$ is the length of $AC$. If $a = 3$, and $b = 4$, we can use Pythagoras' ...
1
vote
3answers
28 views

Sum of segments inside a right triangle.

I am interested for a problem involving the sum of segments inside a right triangle. Consider a right triangle of hypotenuse $\overline{BC}$ and catheti $\overline{AB}$ and $\overline{AC}$. From the ...
2
votes
1answer
75 views

Geometry Problem about tangent lines

Let S be the circumcenter of ABC. $A_0$ is the middle of arc BC not containing A, $C_0$ also the middle of arc AB without C. Let $S_1$ be a circle with center $A_0$, tangent to BC, $S_2$ with center ...
1
vote
0answers
346 views

How to find mass points and ratios in a triangle?

How to find mass points with weights and ratios is my question. In my class, we learned about mass points. First we had the given ratios of 2 side lengths. Given: MC = d MB = e MA = f BD:DA = ...
1
vote
1answer
69 views

An altitude is divided into 5 equal parts by 4 lines. Prove that the the areas of alternate sections are equal.

The question is as follows : Let their be a triangle ABC. Make altitude AD on C. Divide this altitude in 5 equal parts with lines EF, GM, IJ, KL intersecting at points M,N,O,P respectively. We have ...
0
votes
1answer
25 views

Trying to figure out coordinates of isoscleles triangle

I'm programming some application, which heavily depends of geometry. Let's say, in 2D coordinates system I have i.e. : Bxy = (5,-2) Cxy = (2,-5) ABlength = 5.5 ...
-1
votes
1answer
71 views

In a Right Angled Triangle.

In a triangle ABC, Let $\angle$C=$\frac{\pi}{2}$. If $r$ is the inradius and $R$ is the circumradius, then what is the value of $2r+R$. Options are a+b b+c c+a a+b+c My approach. Radius of ...
0
votes
1answer
39 views

How is the area of this triangle calculated

I was reading "Problems of Calculus in one variable" by I A MARON, and came across this solved example in first chapter which I am unable to comprehend, please help me understand this. Scan of the ...
2
votes
1answer
51 views

Find cosine of acute angles in a right triangle.

If sides of a right triangle are in Geometric Progression, then find the cosines of acute angles of the triangle. [Answer] $\frac{\sqrt{5}-1}{2}$,$\sqrt\frac{\sqrt{5}-1}{2}$ My work: Using ...
0
votes
1answer
27 views

Two questions regarding the angle of reflection

I have two problems regarding the calculation of angles given certain values. In the first problem I need to calculate the angle X given that both angles Y are identical In the second problem I ...
0
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2answers
125 views

North has 0 degree and right angle has 90 degree although both are in same position

Before reading trigonometry I guessed that if a line is pointing to north then it has 0 degrees and increases clockwise. But now I see right angle has 90 degree though that is in the same position as ...
0
votes
1answer
54 views

How to find circumference origin position?

I need to find origin of circumference which is defined by two points and vertex angle of isosceles triangle: I've got radius of triangle by ...
1
vote
3answers
66 views

How was this equation for the hypotenuse of a triangle derived?

I've been staring at this for quite a while and simply can't understand how they got the equation for the hypotenuse. Probably has something to do with it being 5am my time! I'm confused because (...
0
votes
1answer
508 views

using slope to find an angle in right angled triangle

I have a right angled triangle in which I know the length and the slope of the hypotenuse, how do I find one of the angles? Thanks
1
vote
1answer
45 views

Coordinates of a vertex of a triangle?

Here is the problem: There is a triangle with vertices $A,B,C$ in a cartesian coordinate system, where coordinates of points $A$ and $B$ and the angle $\alpha=\measuredangle ABC$ are given. The ratio $...
2
votes
2answers
325 views

How many triangles in the picture?

Sorry if this has already been asked before. Is there any formula for such questions? EDIT: I have numbered the smallest triangles in the picture and marked the pentagon as x. Then I listed all ...
2
votes
2answers
44 views

$A,B,C$ satisfy $\sin 2A: \sin 2B: \sin 2C= 5:12:13$ find $A$?

I would appreciate if somebody could help me with the following problem: Question: $A,B,C$ satisfy (1), (2) (1). $A+B+C=\pi(0< A,B,C< \pi)$ (2). $\sin 2A: \sin 2B: \sin 2C= 5:12:13$ Find $A$ ...
0
votes
1answer
39 views

Find length of side

I tried to solve this problem ... but i can't find answer. Anyone can help me? EBC=90 & DCB=90 & AHC=AHB=90
2
votes
1answer
51 views

Prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$

If $x, y, z$ are the sides of a triangle, then prove that $|x^2(y − z) + y^2(z − x) + z^2(x − y)| < xyz.$ This is a self-answered question.
3
votes
2answers
59 views

Proof related to circumcircle of triangle

I have a triangle $ABC$ with incenter $I$. $AI$ extended meets the circumcircle of $ABC$ at $M$. Prove that $CM=BM=IM$. I was able to prove that $CM=BM$ taking advantage of the fact that the ...
8
votes
3answers
772 views

Dividing an obtuse triangle into acute triangles

Can an obtuse triangle be subdivided into only acute triangles (right triangles are not allowed)? Any number of subdivisions can be made as long as all of the angles in all resulting triangles are ...
0
votes
1answer
24 views

In this figure find AC=x

Can you find $AC$, when only the angle $DBC$ and $DEB$ are $90$ grades. I can't because I think they should give the angle $CAB=90$ grades too.
1
vote
1answer
92 views

Euclid I.24 Proof Why is DFG greater than EGF?

Proposition 24 If two triangles have two sides equal to two sides respectively, but have one of the angles contained by the equal straight lines greater than the other, then they also have the ...
2
votes
2answers
244 views

To prove in a triangle: $AD^2=AB\cdot AC- BD\cdot CD$

If $AD$ is an angle bisector of $\triangle ABC$ (with $D\in BC$), then we have to prove that: $$AD^2=AB\cdot AC- BD\cdot CD$$ I have no idea how to do this, can this be proved with simple geometry? :...
0
votes
1answer
328 views

Lengths of the sides of a triangle: sufficient and necessary condition?

For any three positive scales, $a,b,c$, what is the sufficient and necessary condition such that they can form a triangle? Is $a+c>b,a+b>c,b+c>a$ enough? Thanks!
1
vote
0answers
28 views

Circles intersecting at A and B [duplicate]

Question: Two given circles intersect at A and B. A straight line through B meets the circles again at C and D. Prove that CD is greatest when it is parallel to the line joining the centres My ...
3
votes
3answers
450 views

Trigonometry. Finding the angle alpha

Refer the diagram below : What should be the angle alpha such that the variable x is between 7mm and 7.3mm.
0
votes
2answers
111 views

Isosceles triangle and scalene triangle

Question: Given the base and vertical angle of a triangle show that its area is greatest when the triangle is isosceles. My attempt: For isosceles triangle (with base given 2x, and vertical ...
0
votes
1answer
634 views

Can we find out the area of conical frustum by using triangles?

I have been trying to find out the area of conical frustum by using triangles.
13
votes
4answers
1k views

Does a triangle always have a point where each side subtends equal 120° angles?

Is there a point $O$ inside a triangle $\triangle ABC$ (any triangle) such that the angle $\angle{AOB} = \angle{BOC} = \angle{AOC}$? What do we call this point?
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vote
0answers
35 views

Triangles with vertices on conics and their foci

Let $A$, $B$, and $C$ be the lengths of the three sides of a triangle. Let $α$, $β$, and $γ$ be the measures of the angles opposite those three sides respectively. Mollweide's formula tells us that $$...
0
votes
1answer
159 views

applied optimization problem- triangle fence

A farmer is trying to fence off a field on the edge of a river. He has two 1km long sections of fence to use to make a triangular field. The edge by the river does not need fencing, and the fence ...
5
votes
1answer
52 views

Is a triangle with two equal angles always isosceles?

An isosceles triangle is a triangle with two sides that are equal in length. This means that two angle will also be equal to each other. Is there any way that a triangle could have two (only two) ...
0
votes
1answer
31 views

Related rates question.

Two sides of a triangle have lengths $\sqrt{21}~m$ and $\sqrt{7}~m$. The angle between them is increasing at a rate of $\dfrac{2}{\sqrt{3}}~rad/sec$. How fast is the altitude of the triangle ...
2
votes
3answers
84 views

Geometry question involving triangle

Question: $ABC$ is a right angle triangle at $A$. $AD$ is the altitude through A; E is a point on AC such that $AE=CD$. F is a poibnt on AB such that $AF=BD$. Prove that $BE=CF$. Challenge ...
0
votes
3answers
176 views

Getting 90 degree coordinate of 2 coordinates that you know

I have 2 coordinates and I need to find the third with a 90 degree angle. How could I do this? ...
2
votes
4answers
185 views

Geometry question involving triangles given with picture.

Here's the question: $\overset{\Delta}{ABC}$ is a triangle. $D$ is a point on $[BC]$. $|BD|=4$. $|AD|=|CD|$. $\text m(\widehat{CBA})=\alpha=30^\circ$. $\text m(\widehat{ACB})=\beta=...
1
vote
1answer
207 views

How Many Triangles are Created by n Lines in the Plane?

Suppose we are given n lines in the plane in "general position", which in the present case we define to mean the following: A. no 2 lines are parallel or identical B. no 3 lines have common ...
1
vote
1answer
70 views

Area of Triangle

The position vectors of $A$ $B$ and $C$ relative to an origin $O$ are given by $OA=(2,1,3)$ $OB=(0,-1,7)$ and $OC=(2,4,7)$ Part i) Show that angle $BAC= \cos^{-1}(\frac{1}{3})$ Part ii) Using the ...
1
vote
0answers
45 views

Find all the triangles satisfying $\cos(A)\cos(B)+\sin(A)\sin(B)\sin(C)=1$ [duplicate]

I am trying to solve the problem of finding all triangles with angles $A$, $B$ and $C$ (in $[0,\pi]$) such that $\cos A\cos B+\sin A\sin B\sin C=1$. In the case where the triangle has a right angle, ...
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vote
5answers
16k views

How can I find the lenght of the third side of any triangle

I will know the length of two sides of any triangle that I use, but I will not know any of the angles. I know how to find the length of the third side if I knew the angle where I am sitting, but how ...