For questions about properties and applications of triangles

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5
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3answers
232 views

Finding the area of the 4th triangle, given the areas of the other 3, and all the 4 form a rectangle

In one of my tutorial classes, when I was studdying in 9th class (I am in 10th now), our tutor gave us a problem saying it’s a difficult one, and to him, it was incomplete. This is that problem: ...
1
vote
1answer
27 views

Given the incentre of $\Delta ABC$ and the equations of the angle bisectors what is the locus of the centroid of the triangle $ABC$?

I got this problem on a test yesterday Consider $\Delta ABC$ with incenter $I(1,0)$. Equations of the straight lines $AI$, $BI$, and $CI$ are $x=1$, $y+1=x$ and $x+3y=1$ respectively and $\cot \left( ...
0
votes
2answers
60 views

Triangle whose side lengths and area are rational numbers [on hold]

Does there exist a triangle with side lengths given by rational numbers $x$, $2x$, and $y$ such that the triangle's area is also rational number?
16
votes
4answers
2k views

Proving Stewart's theorem without trig

Stewart's theorem states that in the triangle shown below, $$ b^2 m + c^2 n = a (d^2 + mn). $$ Is there any good way to prove this without using any trigonometry? Every proof I can find uses the ...
3
votes
3answers
49 views

Determine length from sketch

I have a simple problem that I need to solve. Given a height (in blue), and an angle (eg: 60-degrees), I need to determine the length of the line in red, based on where the green line ends. The ...
1
vote
1answer
29 views

If $\vec{AA_1} + \vec{BB_1} + \vec{CC_1} = 0$ prove that the triangle is equlateral.

The problem states that if $AA_1, BB_1$ and $CC_1$ are the altitudes of the triangle $\bigtriangleup ABC$ and $\vec{AA_1} + \vec{BB_1} + \vec{CC_1} = 0$ then the triangle is equilateral. My solution: ...
1
vote
2answers
63 views

Getting different answers using different methods in a geometrical problem

Problem statement: Given a triangle with side lengths 4 and 6, their corresponding opposite angles have a 1:2 ratio. Find the length of the third side. I solved the problem in 2 ways and got as an ...
0
votes
1answer
104 views

Help determining angle

Let $R$ be the triangle defined by $−x\tan(\theta) \le y \le x\tan(\theta)$ and $x \le 1$ where theta is an acute angle. Sketch the triangle and calculate \begin{equation*} \iint_R(x^2+y^2)\mathrm ...
1
vote
1answer
46 views

What is the name of this (circumscribed) triangle?

I am meeting the following triangle more and more in my investigations of ideal triangles in the Beltrami Klein model of hyperbolic geometry. That made me wonder: is there a name for it? (And does it ...
1
vote
2answers
34 views

Help for a problem with inscribed triangles

If we have a triangle $ABC$ with $AB = 3\sqrt 7$, $AC = 3$, $\angle{ACB} = \pi/3$, $CL$ is the bisector of angle $ACB$, $CL$ lies on line $CD$ and $D$ is a point of the circumcircle of triangle $ABC$, ...
-1
votes
2answers
334 views

How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
5
votes
3answers
120 views
+50

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 ...
1
vote
4answers
30 views

Inequality for sides and height of right angle triangle

Someone recently posed the question to me for the above, is c+h or a+b greater, without originally the x and y lengths. I used this method: (mainly pythagorus) $a^2+b^2=c^2=(x+y)^2=x^2+y^2+2xy$ ...
3
votes
1answer
78 views

Question about Pasch's Postulate, line going through all three sides of a triangle

I've been reading the textbook Elementary Geometry from an Advanced Standpoint by Edwin E. Moise (3rd ed.). My problem with his wording of Pasch's Postulate, and then a subsequent problem which ...
3
votes
2answers
81 views

How to prove the the addition of tangent is the same as the multiplication? [duplicate]

If A,B,C are angles of a triangle show that: $$\tan A+ \tan B+\tan C = \tan A \tan B \tan C $$ I've tried this many times but I cannot seem to prove it, can someone show me how to solve this ...
3
votes
1answer
41 views

Triangle Center Midpoint

Consider the following construction of a triangle center: (The method could also be easily generalized to any shape with finite perimeter) For each point $X$ on the triangle, find point $X'$ such ...
0
votes
2answers
32 views

Calculating the right angled triangle's cathetus

We just started learning the Pythagorean theorem at school and we got a pretty difficult assignment. 5 meter tall bamboo broke and the top of it touched the floor 2 meters from the base of the ...
2
votes
2answers
26 views

Triangle relationships

I was wondering if someone can help me actually. You see I came upon this book called Mathematics for Physics by Michael and Malcolm Woolfson. I a presently stuck on the very first exercise and I can ...
6
votes
0answers
64 views

Number of ways to dissect a square into triangles of equal area

Monsky's theorem states that it is impossible to dissect a square into an odd number of triangles of equal area. If $n$ is an even integer, I am interested in the number of ways of dissecting a ...
1
vote
1answer
46 views

Ravi substitution in inequalities

There is a well-known substitution for proving geometric inequalities: If $a,b,c$ are the side lengths of a triangle, then in an inequality involving $a,b,c$ it is possible to replace $a,b,c$ by ...
0
votes
1answer
22 views

How to prove Thomsen's theorem?

Thomsen's theorem states that given a triangle ABC, choosing a point on AB (but not A or B) and doing the internal path parallel to AC till reaching BC, and then doing the path parallel to AB till ...
-3
votes
1answer
48 views

Please help with this question about angle of elevation and depression [closed]

A surveyor needs to determine the height of a building. She measures the angle of elevation of the top of the building from the two points, 38cm apart. The surveyor's eye level is 180cm above the ...
2
votes
2answers
38 views

Is there a measure for how thin or squat a triangle is?

Is there a measure for how thin or squat a triangle is? Similar to eccentricity for ellipses.
4
votes
2answers
586 views

Prove that the centre of the nine-point circle lies on the midpoint of the Euler line

In $\Delta ABC$, $AD, BE, CF$ are the altitudes and $\Delta A'B'C'$ is the medial triangle. $K, L, M$ are the midpoints of $AH, CH, BH$. Consider the nine-point circle with centre $G$ (not to be ...
-1
votes
0answers
54 views

Very dificult: Triangle limits. [duplicate]

Obtain the following formulas : The maximum height corresponding to the side b of any triangle (abc) once known the value of its perimeter and height corresponding to the a side. The minimum ...
5
votes
2answers
212 views

Competition style problem circa 1992

We're given a triangle $ABC$. Going clockwise, let $B_1$ and $B_2$ be distinct points on the segment $AC$ ($B_1$ is between $A$ and $B_2$), let $A_1$ and $A_2$ be distinct points on the segment $CB$ ...
2
votes
0answers
30 views

Is there an equidissection of a unit square involving irrational coordinates?

An equidissection of a square is a dissection into non-overlapping triangles of equal area. Monsky's theorem from 1970 states that if a square is equidissected into $n$ triangles, then $n$ is even. ...
2
votes
4answers
31 views

Triangle Inequality with Complex Numbers

I was wondering how to prove the triangle inequality with complex numbers: Verify that the function $d(z_1, z_2)$ is a distance funtion on $\mathbb{C}$ and also on any subdomain on $\mathbb{C}$. I ...
1
vote
1answer
75 views

The minimum perimeter and maximum height of a triangle under constraints

I'm developing a web application that consists of a calculator triangles. Although I am not a mathematician, with paper, derive and Geogebra I managed to get a lot of formulas to calculate a triangle ...
0
votes
2answers
68 views

Hyperbolic Ideal Triangle

I have everything pretty much figured out everything but I need help proving the unique point formed by the three perpendiculars in the picture
0
votes
1answer
33 views

Finding the lengths of this triangle?

please help, i'm to answer the question. The length of AB is 14.67106m. Please give working outs.
0
votes
0answers
22 views

Ideal Triangles and Klein Beltrami Disc

I'm trying to prove something with the ideal triangle in hyperbolic geometry and someone told me that the ideal triangle looks like a euclidean triangle inscribed in a circle in the Klein Beltrami ...
-1
votes
1answer
26 views

Geometry: Perimeter of triangle formed by intersections of tangents

I'm a bit stuck on the question below, and I wondered if anyone out here might be able to help: Construct a circle with a centre in O(0,0) and a radius of 5. Two tangents of the circle intersect in ...
0
votes
1answer
14 views

Finding the radius of excircles from a right angled triangle

Right angled triangles have 3 excircles, I'm struggling to find a formula which gives me the radius of all three excircles, I've been struggling with this for a while. I've done some googling and I ...
1
vote
2answers
61 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 ...
4
votes
3answers
95 views

Prove that this triangle is equilateral?

Given $\triangle ABC$. Let $D$ be the point where the altitude form the $A$ vertex intersect $\overline{BC}$ and the point $E$ is the intersect between the bisector of $\angle ABC$ with ...
4
votes
2answers
120 views

Triangle and its circumcircle

Let $ABC$ be a triangle and $\Gamma$ its circumcircle. On sides $AC$, $BC$ lies respectively points $E$, $F$ such that $CE=BE$ and $CF=AF$. $CM$ is a median of triangle $EFC$. Show that line $CM$ pass ...
15
votes
4answers
213 views

A Triangle Determinant

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
-1
votes
1answer
38 views

Generating a random num from a triangular distribution [duplicate]

http://en.wikipedia.org/wiki/Triangular_distribution#cite_note-1 under "Generating Triangular-distributed random variates" given that U is a number between 0 and 1, what happens if the a, b and c ...
-1
votes
0answers
9 views

Explain how to calculate the third vertex of a 2D equilateral triangle given two other vertices, using trigonometry

Before you comment that this has been asked multiple times, please read further. Given 3 arbitrary points: A(a, b), B(c, d), C(e, f), where AB = AC = BC, find C, knowing the values of A and B, using ...
1
vote
1answer
50 views

Show that 3 circles related to a triangle intersects at common point

We have triangle $ABC$ and points $D,E,F$ which lies repectively at $BC,AC,AB$. There are circles passing through $AFE$, $FDB$, $CDE$ show that they intersect at common point
2
votes
1answer
64 views

How one can show Gerretsen's inequality?

I read from http://rgmia.org/papers/v6n3/wsh.pdf the following: A triangle with semiperimeter $s$, circumradius $R$ and inradius $r$ satisfies $$16Rr-5r^2\leq s^2\leq 4R^2+4Rr+3r^2.$$ How can I prove ...
1
vote
1answer
99 views

How to prove a triangle similarity problem

If I have a triangle $ABC$ with point $E$ lying on $BC$ and point $D$ lying on $AB$ where $AE$ is the height to $BC$ and $CD$ is the height to $AB$, how can I prove that triangle $ABC$ is similar to ...
1
vote
2answers
43 views

Find the acute angles of this right triangle.

I am having trouble finding the acute angles of this triangle. O is the intersection of the medians of the triangle and $OG = \frac{1}{2}OH$. Any suggestions?
2
votes
0answers
17 views

Statue and a flag distances

Next to a flagpole is a statue that measures 9m high. The upper end of the flagpole with the bottom of the statue form an angle of 53.13 degrees to the floor, and the upper end of the flagpole to the ...
1
vote
3answers
53 views

Why are trig functions defined for the unit circle?

Why did we ever need to define the trig functions of angles greater than 90 degrees or less than 0 degrees? What is the use of applying trig functions to such angles? If we apply the trig functions ...
0
votes
2answers
31 views

Bounding inradius, given circumradius.

The problem in my book is as follow. In a $\Delta ABC$ , if $r=r_2+r_3-r_1$ and $\angle A >\dfrac{\pi}{3}$ , then the range of $\dfrac{s}{a}$ is equal to: (Here $r_i $ are exradii) I used ...
0
votes
1answer
40 views

Finding the angle?

I have two circles which share a radius of R units, and each circle contains the center of the other circle. I found that the area of the segment would be, $\theta$ is the central angle between the ...
-1
votes
1answer
27 views

Incenter divide ratio

Given a triangle $ABC$ and angle bisectors $BD,CE$ which intersect at $O$ (incenter) . The ratio in which $O$ divides $BD$ is $3:2$ and it divides $CE$ in ratio $1:2$ . Find the ratio in which the ...
6
votes
2answers
66 views

The conjecture that no triangle has rational sides, medians and altitudes

I have found a conjecture that there is no triangle whose sides, medians, altitudes and area are all rational. I figure that someone must have already found such a triangle if one existed and yet I ...