For questions about properties and applications of triangles

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2
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1answer
53 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 = ...
0
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0answers
41 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.
0
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1answer
32 views

Relationships in a triangle

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

Find the lengths of two segments in a triangle with a line parallel to a side

Take a look at the picture, I am suppose to find the value of x and y. I have already manages to figure it out but here are a few questions that I need to understand. AD/AB = CE/CB = CA/ED x= ...
-1
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0answers
8 views

find the perimeter of the quadrilateral [closed]

Given, perimeter of triangle QST IS 7+ROOT 13. Find the perimeter of PRST?
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0answers
73 views

Do you have any idea?

Let $M$ be a point moving inside a triangle $ABC$ with all sharp angles, with the property that $$\angle(B) + \angle(AMC) = 180.$$ Knowing that $\{E\}:= AM \cap BC$ and that $\{F\}:=CM \cap BA$, ...
-4
votes
1answer
29 views

Which is true: $|x| - |y| < |x-y|$ OR $|x-y| < |x| - |y|$ [closed]

Which is true: (for rational numbers) $$ \lvert x \rvert - \lvert y \rvert < \lvert x - y \rvert $$ or $$ \lvert x-y \rvert < \lvert x \rvert - \lvert y \rvert $$ ? thanks in advance
0
votes
1answer
26 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: ...
0
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1answer
27 views

Find α (Triangles) [closed]

Find $\alpha$ if $A = 4\alpha$. Can someone explain to me how to do this?
2
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1answer
60 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 ...
2
votes
2answers
77 views

Geometry question about centroid

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
16 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
47 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 ...
1
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3answers
38 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
101 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
30 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
vote
1answer
51 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
votes
1answer
18 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 ...
0
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0answers
36 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
votes
3answers
45 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 ...
0
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0answers
22 views

Are isosceles right triangles the only ones whose circumcenters lie on their incircles? [duplicate]

I recently (stupidly) asked this question, to which user Blue responded quickly with the example of the isosceles right triangle. Which triangles have circumcenters on their incenters? Do they have to ...
1
vote
1answer
24 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?
0
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1answer
38 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 = ...
1
vote
4answers
48 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
votes
1answer
16 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 ...
3
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0answers
57 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
vote
1answer
22 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
73 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 ...
2
votes
3answers
62 views

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

The orthocenter of $\triangle ABC$ is $H$. Let $B'$ be a point on AB 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
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0answers
15 views

General formula for n-Simplex side-lenghts given n-volume and angles

Given a flat triangle's three angles $\phi_i $, and its area $A$, you can calculate the $i$th sidelenght $s_i$ (using Einstein's sum-convention) like so: $$ s_i=\frac{\sqrt{2A} \sin \left(\phi ...
0
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0answers
20 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)}
1
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0answers
22 views

Intersections of convex hulls

Given a set of $n$ points $\{A_1, \ldots , A_n\}$ of the plane and every possible triangle formed with $3$ points $A$, I would like to describe the intersections fo theses triangles. By intersection, ...
1
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0answers
16 views

Finding other two vertices when one vertex and each point on the triangle is known ?

I am working on some gesture recognition for my game. I am stuck on a problem. I have one vertex i.e the starting point and every point on the triangle, I also have the centroid. So how do I find the ...
0
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1answer
16 views

Finding other two vertices of a triangle from centroid and one vertex?

I am working on some gesture recognition for my game and I want to find if a point is inside the triangle created by the user or not. For that I need three vertices. Currently I am using the '$1 ...
1
vote
1answer
44 views

Find closest point in triangle given barycentric coordinates outside

Given a non-degenerate triangle ABC and an arbitrary point P in 3D space, I can project P onto the plane defined by ABC and check whether the triangle contains it as described here. I end up with ...
2
votes
1answer
39 views

Minimum Distance between a Triangle and a Distance Field 3D

I am looking for (possibly numerical) solution to this geometric problem: Given a filled 3D triangle $T = \text{conv}(p_1, p_2, p_3) \subseteq R^3$, and a distance field $D(x) : R^3 \to R$, what ...
10
votes
1answer
93 views

Is there a name for the recursive incenter of the contact triangle?

Recently, I became aware that there are many more triangle centers than the four I learned about in school. This reminded me of a thought I had when I first learned about the incenter: what point ...
3
votes
1answer
41 views

Closest point on a 3D triangle, is this algorithm correct?

Given a point $P$ and three triangle vertices $U$, $V$, $W$, all in $\mathbb{R}^3$, I need to find the point in the triangle $UVW$closest to $P$. Does the following algorithm work, or have I missed ...
3
votes
2answers
79 views

Why is the volume one third of that? I mean, where's the fault in my logic? [duplicate]

The volume of a cuboid is $l \times b \times h$. That is, it is equal to base area times height. I think it means that the base is added up height times, it becomes volume (It makes sense if we think ...
2
votes
3answers
76 views

Coordinate of the excentre of a triangle

I am just wondering that how the coordinate of the excentre comes out if we know the coordinates of vertices of the triangle.
0
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2answers
41 views

Proving that the orthocenter lies on $OD$?

While trying to solve this question using GeoGebra, I realized the following curious thing: If $I$ is the incenter of $\triangle ABC$, $ID \perp BC$ with $D$ on $BC$, $AD \perp IO$ with $O$ on ...
3
votes
2answers
65 views

Find the angle if the area of the two triangles are equal?

Let $I$ be the incenter of $\triangle ABC$, and $D$, $E$ be the midpoints of $AB$, $AC$ respectively. If $DI$ meets $AC$ at $H$ and $EI$ meets $AB$ at $G$, then find $\angle A$ if the areas of ...
0
votes
3answers
53 views

Sum with three notations around it.

Have seen the below notation (How to calculate number of triangles and points after dividing a triangle n times?) and need to break it down into plain english so to speak. This just so I can catch up. ...
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1answer
73 views

How to calculate number of triangles and points after dividing a triangle n times?

When having a triangle and dividing it n times, how to get the number of triangles and points? ...
1
vote
1answer
24 views

Solution of triangles

Find the angle at the vertex of an isosceles triangle of maximum area for the given length 'l' of the median to one of its equal sides. I tried to get a relation between l and one of the equal sides ...
1
vote
2answers
35 views

Finding the area of a triangle, given the distance between center of incircle and circumscribed circle

Consider the following depiction: $ABC$ is an isosceles triangle ($AB=AC$), where the two angles opposite the equal sides are equal $\beta$ ($\beta>60$), and $AD$ perpendicular to $BC$. $O$ is ...
1
vote
2answers
54 views

Minimise the Sum of the Areas of the Circumcircles

In a triangle $ABC$, point $X$ is picked on $BC$ such that the sum of the areas of the circumcircles of $ABX$ and $ACX$ is minimised. Describe where $X$ would be located on $BC$, and prove that this ...
1
vote
3answers
49 views

How to determine the equation and length of this curve consistently formed by the intersection of Circles

Consider a Point $A$ that moves linearly on the positive $x$-axis with the velocity $1$ m/s and another Point $B$ at a distance $L$ from $A$ with position $(L,0)$. With each forward motion of point ...
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votes
5answers
82 views

Area of triangle with given coordinates of the vertices

The question for my math is: "Sharon made a scale drawing of a triangular park. The coordinate for the vertices of the park are: $(-10,5)$, $(15,5)$, $(10,12)$. What is the area of the triangular ...
16
votes
1answer
200 views

Under what conditions will the rectangle of the Japanese theorem be a square?

In geometry, the Japanese theorem for cyclic quadrilaterals states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle. Question. Under ...