For questions about properties and applications of triangles

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2answers
56 views

Trignometric inequality

if $\alpha, \beta, \gamma$ are angles of a triangle. prove that $\csc(\frac\alpha2)+\csc(\frac \beta2)+\csc(\frac \gamma2) \ge 6$. I started from $\alpha + \beta + \gamma = 180^{\circ}$ and then I ...
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3answers
84 views

Calculate the length of the sides of a triangle from the area and the angle

I need to find the length of the sides of a triangle. I have an angle and the area of the triangle. I have the answer but I don't know how to figure it out so it doesn't help. The area of the ...
3
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3answers
152 views

proving a median in a triangle base on center of mass

I was wondering: I know that the center of mass in a triangle is divided the medians in the ratio of $2:1$. Is the opposite is true? I mean, if i have triangle $ABC$ and point $D$ is on $BC$, point ...
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1answer
51 views

Finding Triangle's Angle

Find the value of $x$ if the area is $42\;\mathrm{cm}^2$. The triangle is not right angled or an isosceles. $X$ is the angle in the bottom left corner of the triangle that I need to find. The left ...
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2answers
65 views

Let $a,b,c$ be the lenghts of the sides of a triangle. Suppose that $ab+bc+ca=1$. Show that $(a+1)(b+1)(c+1)<4$

Let $a,b,c$ be the lenghts of the sides of a triangle. Suppose that $ab+bc+ca=1$. Show that $(a+1)(b+1)(c+1)<4$. My attempt: I tried multiplying the whole thing but that didn't help at all. ...
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2answers
172 views

Diagonal line through rectangle always creates two congruent triangles (?)

1) Is it true that the diagonal line through a rectangle always creates to congruent triangles? 2) If a quadrilateral has two right angles that are opposite (is this the right word to use), as ...
2
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4answers
98 views

In a triangle, prove that $a\cos A+b\cos B+c\cos C=\frac{8\Delta^2}{abc}$

I have to prove that for a triangle, $$a\cos A+b\cos B+c\cos C=\frac{8\Delta^2}{abc}$$ where $a,b,c$ are the lengths of the sides opposite to the angles A,B,C respectively. I followed the following ...
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0answers
21 views

Length of a right triangle's hypoteneuse projected onto a sphere

Please forgive me if this is the wrong kind of question, but I need someone to verify or refute my work. One leg of a triangle has length, $b$ (base), resulting from angle theta swept out by a ray ...
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5answers
266 views

Find the length of a leg of a right triangle, given the area and the length of the other leg

The length of one leg of a right triangle is $(x - 6)$ centimeters, and the area is $(\frac12 x^2 - 7x + 24)$ square centimeters. What is the length of the other leg? I think the equation that I need ...
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2answers
38 views

How to find the remaining segment of this triangle without the Law of Cosines?

I am given a triangle $T$ with vertices $A, B,$ and $C$ and the the following info about $T:$ $\overline{AC} = 6$ Km $\overline{BC} = 9$ Km The angle formed by these two segments is $120^{\circ}$ ...
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1answer
48 views

Area of an isosceles triangle where the tangents of some angles are in geometric progression

In $\triangle ABC$, $AB=BC$ and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $\overline{BE}=10$. The values of $\tan CBE$, $\tan DBE$, and $\tan ABE$ form ...
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2answers
42 views

Solutions of triangles - proof

Question: For a triangle ABC, prove that: $$r_1 + r_2 + r_3 = r + 4R$$ Where $r_1,r_2,r_3$ represent the radius of the ex-circles opposite to angle A, B, and C respectively. $r$ represents the ...
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1answer
58 views

How to solve a triangle knowing just two legs?

Given a right triangle, if I know two of its legs are $a = 3,5$ and $b = 5,5$, is it possible to solve the triangle with this info? This is, can I determine all of its sides and all of its angles with ...
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0answers
47 views

Finding the coordinates of the third point in a triangle using simultaneous equations

I have 3 circles A,B & C that touch each other at tangents. The centre points of these 3 circles are to be joined to create a triangle. I know the coordinates of 2 of the circles centre points ...
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1answer
91 views

Triangles in geometry please help [closed]

find the length of the altitude to the hypotenuse of a right triangle whose sides have lengths 6.8 and 10 smaller part 3.8
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3answers
219 views

Prove that the circumcenter of $\triangle PIQ$ is on the hypotenuse $AC$.

In right angled $\triangle ABC$ with $\angle B=90 ^{\circ}$, $BD$ is an altitude on $AC$. $P,Q,I$ are the incenters of $\triangle ABD,\triangle CBD$ and $\triangle ABC$ respectively. Prove that the ...
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3answers
4k views

Find the length of the chord given that the circle's diameter and the subtended angle

A chord of a circle subtends an angle of 89 degrees at its centre. Find the length of the chord given that the circle's diameter is 11.4 cm. The problem I have here is that I can't visualise this ...
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1answer
35 views

Geometry: Perpendicular tangent

I came up with this but I have not been able to solve it. I would really appreciate any help. Let $ABC$ be a triangle and let $\omega$ be its circumcircle. Produce the internal angle bisector of ...
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1answer
58 views

Proving that $KL$ bisects $AJ$ in a triangle?

Let $ABC$ be an acute triangle, and its incircle touch the sides $AB$ and $AC$ at $K$ and $L$. Let $J$ be the incenter of $\triangle BCD$, where $D$ is a point on $AC$ such that $BD=AB$. Prove that ...
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1answer
43 views

triangle circle inside it, heights prove exercise

Point $O$ lie inside $ABC$ triangle. Points $A1,B1,C1$ are projections of $O$ on heights led from $A,B,C$ Prove that if $AA1=BB1=CC1$ then $AA1=2r$, where $r$ is radius of circle inscribed in $ABC$ ...
3
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1answer
79 views

Construction of a triangle given some special points ($O,H,I$)

I'm a newbie in this site. I tried to search if this question was already answered but I'm not sure on how to do it. The problem is: given three distincts points $O,H,I$ namely the circumcenter, the ...
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2answers
193 views

Trisecting the sides of a triangle.

Consider the hexagon formed by the six points which trisect the sides of a triangle(two on each side). Is is true that when we connect opposite points in this hexagon, the lines intersect at a single ...
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3answers
46 views

Find the area of the triangle under certain preconditions

With vertices $(0, 0)$, $(b, a)$, $(x, y)$, prove the area of this triangle is $\frac{|by - ax|}{2}$. We know area of a triangle = $\frac{rh}{2}$. ($r$ is the base.) Well, we have $r =$ the ...
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1answer
166 views

Finding third vertexes of any triangle where 2 vertex known and all sides length known

I am working with a CAD engine in the head but i working on code only. I have a rectangular tube that need to be put at an angle. I so have the diagonal of the tube where it has to start and stop but ...
2
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1answer
84 views

Calculationg the angle of a triangle

I am trying to find a specified angle of a triangle. In triangle $ABC$, $\angle A = 20^\circ$. $D$ and $E$ are points on $AB$ and $AC$, where $AB=AC$. $\angle EBC = 50^\circ$ and $\angle DCB = ...
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0answers
42 views

Find the angle in a triangle [duplicate]

Find the angle $a$: I came up with 20 degrees but not sure. Can somebody help here.
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1answer
36 views

Relationships in a triangle

Here is the question, I can''t figure out how to explain this algebraically.
0
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1answer
381 views

Z coordinates of 3rd point (vertex) of a right triangle given all data in 3D

this is my first post.. I hope this good I have 1 triangle in space (3D)... and I know all data except the coordinates of 3er point(vertex)... for example this: then: ...
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1answer
39 views

Find α (Triangles) [closed]

Find $\alpha$ if $A = 4\alpha$. Can someone explain to me how to do this?
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1answer
100 views

Prove that the intersection of $BM$ and $CN$ is on the circumcircle of triangle $ABC.$

Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB$ = $\angle BCA$ and $\angle CAQ = \angle ABC$.Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such ...
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2answers
262 views

Geometry question about centroid [closed]

How do you solve this geometry question? In triangle $ABC$ the centroid is $G$ and $D$ is themidpoint of $CA$. The line through $G$ parallel to $BC$ meets $AB$ at $E$. Prove that $\angle ...
0
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1answer
261 views

Triangle Area Ratio Theorem Problems?

Having a hell of a lot of issues with these problems, supposed to be on the topic of triangle area ratio theorem (ratio area of triangles = ratio of triangles' heights x ratio of triangles' bases.) ...
1
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1answer
96 views

Proof of a set of triangles and unit squares

Suppose that there is $S$, a finite set of unit squares. So, $S$ is chosen from a larger grid of unit squares. The unit squares of $S$ are tiled with isoceles right triangles. Each of these triangles ...
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3answers
59 views

Proving that $BI$, $AE$ and $CF$ are concurrent?

Let $ABC$ be a triangle, and $BD$ be the angle bisector of $\angle B$. Let $DF$ and $DE$ be altitudes of $\triangle ADB$ and $\triangle CDB$ respectively, and $BI$ is an altitude of $\triangle ...
5
votes
1answer
119 views

Inequalites of triangle side with $abc = 1$

Let $a,b,c$ be the sides of a triangle with $abc=1$. Prove that $$ \frac{\sqrt{b+c−a}}{a} + \frac{\sqrt{c+a-b}}{b} + \frac{\sqrt{a+b−c}}{c} \ge a+b+c $$
0
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1answer
47 views

How prove that $AB>AC$ in triangle $ABC$?

Point $D$ is chosen inside $\triangle ABC$, and point $E$ on segment $BD$ such that $BD=CE$. Suppose $\angle ABD=\angle ECD=10^{\circ}$, $\angle BAD=40^{\circ}$, and $\angle CED=60^{\circ}$.How prove ...
1
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1answer
78 views

Proving $B$, $C$, $D'$ and $E'$ to be concyclic iff $AB+AC=3BC$?

Let $ABC$ be a triangle with incenter $I$. The incircle of $ABC$ touches $AC$ at $D$ and $AB$ at $E$. Let $DD'$ and $EE'$ be the diameters of the incircle. Prove that $B$, $C$, $D'$ and $E'$ are ...
2
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1answer
34 views

Triangle Inequality with Vectors

If the magnitudes of vectors $\mathbf{a}$ and $\mathbf{b}$ are $5$ and $12$, respectively, then the magnitude of vector $(\mathbf{b-a})$ could NOT be (A) 5 (B) 7 (C) 10 (D) 12 (E) 17 The triangle ...
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1answer
89 views

Triangle $ABC$ and equilateral triangles $ABC'$, $BCA'$ and $ACB'$.

We consider a triangle $ABC$ whose angles are less then $120°$ and construct the equilateral triangles $ABC'$, $BCA'$ and $ACB'$, exterior to $ABC$. $I$ denotes the intersection of $(AA')$ and ...
2
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3answers
68 views

Given an equilateral triangle, show that $MA + MC = MB$.

I have to solve the following problem: Consider an equilateral triangle $ABC$ and $\mathcal{C}$ its circumscribed circle. Let $M$ be a point located on the arc of the circle defined by $[AC]$ which ...
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1answer
29 views

How many points $P$ such that $\angle APB=\angle BPC=\angle CPA $ are there?

Given that $\triangle ABC$ is arbitrary. How many points $P$ such that $\angle APB=\angle BPC=\angle CPA $ are there?
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1answer
48 views

Triangle Identity leads to another Euclidean parallel.

Referring to TriangleIdentity by 伍柒貳 a while ago, considering $\bigtriangleup$ ABC, it is proved that: $$\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C (1*) $$ I want to take angle $A = ...
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4answers
69 views

Proving $\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C$

As the title, By considering $\bigtriangleup$ABC, Prove $$\sin^2A \equiv \cos^2B + \cos^2C + 2 \cos A\cos B\cos C$$ Thanks
0
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1answer
420 views

How to find heading angle to an object who's x,y coordinates are known?

Scenario: I have a map with a marked location on it. I know my x,y coordinates on the map (top left corner is 0,0), my distance from that marked location, my heading angle relative to true north (0 is ...
2
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0answers
69 views

Proving that $AB = AC$. [closed]

In a $\triangle ABC$, $D$ is a point on $BC$ such that $AB+BD=AC+CD$.Let the centroids of $\triangle$s $ABD$ and $ACD$, vertices $B$ and $C$ lie on a circle.Prove that $AB = AC$.
1
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1answer
29 views

Find point in right triangle with given one vector and one point

I am developing a game where the user move a car with his finger. The car is represented as vector (one point and angle of rotation in the screen). When the user start to dragging the car he generate ...
2
votes
2answers
221 views

Prove that $\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}$ for angles of a triangle

Let the three internal angles of a triangle are $a,b,c$. Prove that $$\cos(2a) + \cos(2b) + \cos(2c) \geq -\frac{3}{2}.$$ I'm looking for an elementary, geometric proof. So avoid derivatives and ...
4
votes
1answer
179 views

Nine-point circle equivalent for tetrahedrons?

Nine-point circle for a triangle is defined as the circle that passes through: the midpoint of each side the foot of each altitude the midpoint of the line segment from each vertex to the ...
2
votes
3answers
120 views

Proving $HH', BB', CC'$ are concurrent?

The orthocenter of $\triangle ABC$ is $H$. Let $B'$ be a point on AC and $C'$ be a point on AB, such that $BCB'C'$ is a cyclic quadrilateral. Let the orthocenter of $\triangle AB'C'$ be $H'$. ...
0
votes
1answer
54 views

ABC is a triangle, D is a point in the triangle. E is the midpoint of BD. AB=BC, angle ABD= angle DBC=35 degrees, angle ACD=25 degrees. Angle BAE=?

I tried to solve this problem but couldn't. I just know that here, angle BDC= 100 degrees, angle BAC= 40 degrees, AB^2+AD^2=2(AE^2+BE^2) and AB/AD={sin(angle DAE)}/{sin(angle BAE)}