For questions about properties and applications of triangles

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2
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0answers
55 views

How prove that $II^{\prime}< AA^{\prime}$ for $I $ and $I^{\prime}$ be their incenters?

Assume that we have two triangles $ABC$ and $A^{\prime}BC$. Let $I $ and $I^{\prime}$ be their incenters. How prove that $II^{\prime}< AA^{\prime}$? I have no idea how to do this, can this be ...
3
votes
1answer
108 views

Inequality of length of side of triangle

For any triangle with sides a,b,c $$a^2b(a-b)+b^2c(b-c)+c^2a(c-a)\ge 0$$ I tried substituting $a=x+y$, $b=y+z$, $c=z+x$ but well it doesn't help in any sense except wasting 3 pages that lead to ...
0
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2answers
103 views

How to find the inradius of a triangle with given side lengths?

I need to find the inradius of a triangle with side lengths of $20$, $26$, and $24$. I know the semiperimeter is $35$, but how do I find the area without knowing the height? Thank you.
1
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3answers
161 views

How can I find the sine or the tan or the cos of an angle in radian?

There is an angle equal 0.54 radians and opposite leg equal to 3 units, I need to find the length of the adjacent leg. I know that I have to do ${\rm leg} = \frac{3}{\tan(0.54 \text{ rad})}$. I got ...
1
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3answers
119 views

The area of intersection of an isosceles triangle with another triangle

I tried graphing the equations that form the two isosceles triangles and integrating the bounded area and got 7.456 as my answer after rounding. The answer key has the answer listed as 7.2 However, ...
0
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2answers
30 views

Where to put angle ending on right triangle, only using variables.

Let's say I have a triangle ABC, with side lengths abc. I need to draw a line from the angle connecting the base (c) and hypotenuse (b). I don't know the real angle, but I know it's sin-1. I need to ...
2
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0answers
76 views

Looking for an existing proof for a property of triangles

In my paper, I need the following lemma. I can prove it, but it is a little lengthy to be put inside the paper. I am wondering is there any existing proof that I can quote. Lemma 1: Let the nodes ...
5
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2answers
129 views

Three circles having centres on the three sides of a triangle

NOTE: I would appreciate it if you provided a hint and not the whole solution. BdMO 2014 Nationals: In acute angled triangle ABC, considering a portion of side BC as diameter a circle is drawn ...
0
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1answer
40 views

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A.

In the three angles, A, B, C of a triangle, angle B exceeds twice angle A by 15 degrees. Express the measure of angle C in terms of angle A. I know it looks simple, but my reasoning does not agree ...
0
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1answer
69 views

Trying to prove concurrence of altitudes of a triangle.

I know that this question had been asked before, but I am not exactly following what the answers say. Doing my own way here: I am puuzzled how to continue? I named the points A,B,C, and the foot of ...
0
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1answer
67 views

who discovered the orthocenter of a triangle?

I tried to answer Is there a name for this result in planar geometry? and wanted to go back to the first mention of the orthocenter (or even the altitude of a triangle, but i did draw a complete ...
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0answers
41 views

What's wrong with my reasoning while setting up a limit?

I was writing an answer to this question, which asks about what happens to the apex of an isosceles triangle if a vertex is at infinity. I thought it would be very easy to prove it by setting up a ...
4
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4answers
211 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
147 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
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0answers
32 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
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4answers
60 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 ...
2
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0answers
44 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) + ...
0
votes
1answer
21 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 ...
0
votes
2answers
43 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
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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 ...
2
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1answer
49 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 ...
0
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2answers
54 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 ...
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3answers
24 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
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1answer
59 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
105 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
39 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
23 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 ...
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1answer
63 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
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1answer
32 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
42 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
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1answer
24 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
38 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
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1answer
32 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 ...
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3answers
60 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
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0answers
11 views

Finding the ratio of division by circumcenter

In an acute angled triangle ABC where O is the circumcenter Prove that $BD : DC = sin2C:sin2B \quad$ where D is the point of intersection of AO (extended) with BC. $AO : OD = sin2C + sin2B : ...
0
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1answer
88 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
0
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1answer
28 views

$1,2,3$ task — calculate tangents

Given is square $ABCD$. Point $E$ is the midpoint of segment $CD$ ($E\in CD \wedge |DE|=|EC|$). Point $P$ is common point of diagonal $AC$ and line segment $BE$. ($\lbrace P\rbrace = AC \cap BE$). ...
0
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1answer
31 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
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2answers
172 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
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2answers
41 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
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1answer
36 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
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1answer
46 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
35 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
172 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
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1answer
23 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
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1answer
51 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
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1answer
75 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
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1answer
47 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!
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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 ...
4
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3answers
206 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.