For questions about properties and applications of triangles

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0answers
16 views

Counting number of points making angle < 90

I have a around 1000 points and 1000 segments in the form of $(x_1, y_1, x_2, y_2)$ meaning the segment starts at coordinate $(x_1, y_1)$ and finishes at $(x_2, y_2)$. For each line i want to know how ...
-3
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1answer
40 views

Can you solve this geometric question on triangles? [on hold]

In a triangle $ABC$, $D$ is a point on the side $BC$.Given: $AD=10$,$BD=DC=8$ and $BC*AD=6$.What is the length of $BC$? a.$5$ b.$10$ c.$15$ d.$20$ That was asked in a newspaper quiz.
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4answers
53 views

Triangle - Trapezoid [Geometry]

I'm having trouble with following assignment: "Sides of triangle are $13$, $14$, and $15$. Line parallel to the longest side cuts through the triangle and forms a trapezoid which has perimeter of $39$...
3
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3answers
63 views

Prove: In a Triangle, $II_1 = a\cdot \sec \frac{A}{2}$

Prove that $II_1 = a\cdot \sec \dfrac{A}{2}$. $I$ is center of incircle, $I_1$ is center of excircle. What I did is : Drop $ID \perp AB$, & $I_1F \perp AF$ at $F$ So $ID\parallel I_1F$ $\dfrac{...
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2answers
22 views

Get second vertex of isosceles triangle [on hold]

Given the equal sides of the triangle and the angle $\theta$ between them as well as the other 2 vertices of the triangle how do I get the second base vertex coordinates. Sorry for my poor drawing. <...
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3answers
48 views

Find third coordinate for a right triangle with 45degree angles

I have a right triangle with two 45degree angles. I know the points for the two coordinates opposite the right angle. I need to calculate the missing point. I have seen similar questions here, but ...
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0answers
35 views

Proving Gerretsen's Inequality

Today in class we were shown Gerretsen's inequality: $$16Rr-5r^2\leq s^2 \leq 4R^2+4Rr+3r^2$$ Where $R$, $r$, and $s$ are the circumradius, in radius, and semiperiter of a triangle. After some ...
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2answers
36 views

Prove: $\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$, for circumradius R, inradius $r$, and exradii $r_x$ [on hold]

In $\triangle ABC$, prove: $$\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$$ for circumradius $R$, inradius $r$, and exradii $r_a$, $r_b$, $r_c$ in the standard ...
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1answer
24 views

Geometric proof for properties of Farey sequence

Let $P=(a,c)$ and $P^{'}=(b,d)$ be integral co-ordinates such that $\frac{c}{a}$ and $\frac{d}{b}$ are consecutive terms of Farey sequence. If $O$ is the origin how do I prove no integral co-ordinate ...
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2answers
51 views

Finding the third side of a triangle given the area

I know the area and the lengths of two sides (a and b) of a non-right triangle. I also know P1 (vertex between a and c) and P2 (vertex between a and b). I already know this much: Perimeter = $ \frac{...
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1answer
24 views

Area of all triangles involved in a big triangle.

I have a big triangle made up of several small triangle as depicted in picture given below. Suppose, there is one generic triangle of this shape which is formed by joining points arranged in n rows....
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0answers
24 views

Sum of Area of Circles. [duplicate]

A circle of radius x cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that they are each tangent to two sides of the triangle and to ...
3
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2answers
454 views

Barycentric coordinates of a triangle

I have to do what described in the picture below. Consider the planar triangle $[p_1,p_2,p_3]$ with vertices $p_1=\begin{pmatrix}-2\\-1\end{pmatrix}$, $p_2=\begin{pmatrix}3\\-1\end{pmatrix}$...
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1answer
15 views

Find equal side lengths for isosceles triangle from middle angle and area?

I know that this is a really easy question, but I am looking for the answer to this question: The area of this isosceles triangle is 5cm squared. The angle ABC is 22 degrees. Work out ...
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1answer
37 views

Problem on circles, tangents and triangles

Let $c_1,c_2,c_3$ be three circles of unit radius touching each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed ...
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1answer
433 views

Altitudes Ratio

If h, h', h'' denote the lengths of the three altitudes of a triangle, which of the following ratios never occurs as the ratio h: h': h''? ...
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3answers
30 views

Prove that $AH^2+BC^2=4AO^2$

Prove that $AH^2+BC^2=4AO^2$, where $O$ is the circumcentre and $H$ is the orthocentre of the triangle $ABC$.
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0answers
34 views

How to solve this $80^\circ$-$80^\circ$-$20^\circ$ triangle ($60^\circ+20^\circ$ and $70^\circ+10^\circ$ variant)? [duplicate]

A friend of mine asked me for help with a math problem and I struggled with this for over an hour. I told him sorry, and I felt bad. It's been bugging me now for hours. I don't even so much care for ...
4
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1answer
56 views

Find the sides of the triangle.

The triangle with sides $8-15-13$ has a $60^{\circ}$ angle. The triangle with sides $11-35-31$ also has a $60^{\circ}$ angle. Find a triangle $x-y-403$ where $x$ and $y$ are relatively prime positive ...
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1answer
15 views

Geometry (ratio of subdivided length in a triangle)

In triangle ABC, label X on AB and Y on AC such that AX : XB = CY : YA = 2 : 1. Extend XY and BC such that they meet at point Z. Find ZB : ZC.
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1answer
31 views

Finding area of triangles [closed]

In a triangle, the average of any two sides is $6 cm$ more than half of the third side , then find the area of the triangle (in$\ cm ^ {2}$)
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2answers
33 views

Finding Area of the Triangle [closed]

In the figure, the ratio of AD to DC is 3 to 2. If area of $\Delta ABC$ is 40 $cm ^ {2}$ , what is the area of $\Delta BDC $
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3answers
61 views

If $a^2 + b^2 = c^2$, then $a^3 + b^3 < c^3$, for $a$, $b$, $c$ the sides of a triangle

If $a$, $b$, $c$ are the sides of a triangle where $a^2 + b^2 = c^2$, prove that $a^3 + b^3 < c^3$. I've tried triangle inequality, but I am stuck.
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2answers
54 views

How to find the tangency condition for this circle geometry problem?

Suppose I have a circle $C$ of radius $1$, and I have a chord of this circle, of given length $l$. The chord makes a known angle $\theta$ with the tangent to the circle. I position a smaller circle $...
3
votes
2answers
128 views

If the sides of a triangle satisfy $(a-c)(a+c)^2+bc(a+c)=ab^2$, and if one angle is $48^\circ$, then find the other angles.

In triangle $ABC$ one angle of which is $48^{\circ}$, length of the sides satisfy the equality: $$(a-c)(a+c)^2+bc(a+c)=ab^2$$ Find the value in degrees the other two angles of the triangle. I ...
2
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3answers
172 views

Prove triangle similiarity by given expression

I am working on the following problem, but I can't seem to figure it out. The length of the sides in the triangle $T_1$ are $a_1$, $b_1$ and $c_1$. The length of the sides in the triangle $T_2$ ...
6
votes
2answers
166 views

Triangle in perspective to a given triangle but similar to another

Is it always possible to construct a triangle that is in perspective to a given triangle and have it also be similar to a different given triangle? If you create a triangle in perspective to another, ...
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2answers
57 views

Construct the triangle with given points and lines

On the following picture you see the excersice handed to us. Construct triangle ABC when you know that x is the line that contains points B and C, line z is the median that goes trough point A and ...
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3answers
103 views

Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$

If $A + B + C = \pi$, then find the minimum value of $\cot^2 A +\cot^2 B + \cot^2 C$. I don't know how to solve it. And can you please mention the used formulas first. What I can see is that if one ...
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2answers
133 views

Finding the radius of the smallest circle that can circumscribe an equilateral triangle

Q:A puzzle board is in the form of an equilateral triangle that has an area of $7\sqrt{3}$ if the board is placed on a circular table, what should be the min area of the table so that the whole board ...
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1answer
24 views

3D Geometry concurrency problem

$ABCD$ is a tetrahedron. Let $K$ be the center of the incircle of $CBD$. Let $M$ be the center of the incircle of $ABD$. Let $L$ be the centroid of $DAC$. Let $N$ be the centroid of $BAC$. Suppose $...
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5answers
206 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. I can see that if $A=B=C=\frac\pi3$ then I get $\sin A+\sin ...
0
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1answer
96 views

Hensen inequality in trigonometry: $\sin A + \sin B + \sin C \leq \frac{3}{2} \cdot \sqrt[2]{3} $ [duplicate]

Can anyone help me how to prove $\sin A + \sin B + \sin C \leq \frac{3}{2} \cdot \sqrt[2]{3} $ I have idea use Jensen but how to use it here?
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3answers
58 views

For $\triangle ABC$, prove $( \sin A + \sin B )( \sin B +\sin C )( \sin C + \sin A) > \sin A \sin B \sin C$

In $\triangle$ ABC, prove that $$( \sin A + \sin B )( \sin B + \sin C )( \sin C + \sin A) > \sin A \sin B \sin C$$ I have tried the formula A.M.- G.M. relation with $\sin A$, $\sin B$, and $\...
0
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1answer
536 views

How to find heading angle to an object whose 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 ...
0
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1answer
446 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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1answer
27 views

Triangle wave equation

I have a triangle wave equation represented as $$ y = \dfrac{A \cdot \left(P - \lvert\;\left(x \mod (2 \cdot P) \right) - P \;\rvert\right)}{P} $$ where $A$ is the amplitude and $P$ is half of the ...
2
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1answer
41 views

How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - \cot A)$?

Consider all triangles $ABC$ where $A < B < C \leq \frac{\pi}{2}$. How find the least value of the expression: $M = \cot^2 A + \cot^2 B + \cot^2 C + 2(\cot A - \cot B)(\cot B - \cot C)(\cot C - ...
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1answer
59 views

In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$ [duplicate]

In a $\triangle ABC,$ Evaluation of minimum value of $\cot^2 A+\cot^2 B+\cot^2 C$, Given $A+B+C = \pi$ $\bf{My\; Try::}$ Using $\bf{A.M\geq G.M}$ $$\frac{\cot^2 A+\cot^2 B}{2}\geq \cot A\cdot \...
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0answers
36 views

split a rectangle with triangles into polygons as uniformly as possible

Given a rectangle $A$ and $n$ triangles $\{B_1,B_2,...,B_n\}$, I put the triangles inside $A$, at least one vertex of each triangle is not outside $A$ (inside $A$ or on the edge of $A$). So that A is ...
10
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0answers
2k views

Triangle dissection, no shared edges

It's possible to divide a triangle into smaller triangles such that no edge lengths are shared. Alternately, no two internal triangles share two vertices. The top three are the known simplest ...
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0answers
34 views

Discovering length of line

I'm attempting to work out length of BD from below diagram : The length of BD is -2 +- some value. But since I do not know the y co-ordinate of B can the length of BD be determined from ...
0
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1answer
38 views

Working out length of side of triangle?

I'm taking mooculus course from https://mooculus.osu.edu/exercises/linearTriangles1 and am given following problem : What is the intuition of the hint : 'length of DA = abscissa of D minus abscissa ...
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1answer
466 views

For which $N$ is it possible to alter one side of an equilateral triangle of side length $N$ to get another triangle of integer side lengths, …?

This question is "inspired" by the Rupsa and Equilateral Triangle problem from Code Chef's "October Challenge 2015". The deadline of 12 October 2015 has passed. Given an equilateral triangle having ...
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1answer
39 views

Prove concurrency in a triangle

If a circumference cuts a triangle $ABC$ at its sides $BC$, $CA$ and $AB$ at points $P, P'; Q, Q'; R, R'$; respectively (so twice on each side, and if $AP, BQ$ and $CR$ are concurrent (intersect at a ...
0
votes
2answers
108 views

What is the size of the angle $\angle AMC$? [duplicate]

Suppose we have a triangle $\triangle ABC$ where the size of two angles are given: $\angle B=15^\circ$ and $\angle C=30^\circ$. We draw the median $AM$, so now what is the size of angle $\angle AMC$? ...
0
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2answers
51 views

Angle Between Two Tangents

In the picture below, the angle $AOB$ is $\delta \theta$, and then it is deduced that the angle between the two tangents is the same from the fact that the angles in a quadrilateral add up to $2 \pi$. ...
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1answer
34 views

Question from triangles [closed]

in 🔺ABC, P and Q are points on sides AB and AC respectively, such that PQ||BC . If AP=2.4 cm, AQ=2cm , QC=3 cm and BC=6cm , find AB and PQ?
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4answers
90 views

find the measure of $AMC$

if $M$ is the midpoint of $BC$ then find the measure of $AMC$. I tried to use the angles to find $AMC$ but I don't know how to use that $M$ is the midpoint of $BC$.
1
vote
1answer
39 views

Derive a relation between angles A,B and C

Derive a relation between angles A,B and C (do not use other angles in the final relation): I have tried to use two theorems in triangles(external angle and complement angles),but no success! It ...