For questions about properties and applications of triangles

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8
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2answers
89 views

Area of the given triangle

Through an arbitrary point lying inside a triangle, three straight lines parallel to its sides are drawn. These lines divide the triangle into six parts, three of which are triangles. If the areas of ...
2
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2answers
150 views

Angle bisector divides the triangle into two triangles. Find the area of one of them.

In $\triangle ABC, AB = 12, AC = 10$. $I$ is incenter $∠BIC = 105 ^{\circ}$. Find area of $\triangle ABD$ where $AD$ is angle bisector. I've drawn the following figure: Now, $∠IBD + ∠ICB =75 ...
-1
votes
1answer
330 views

If an equilateral triangle has an area of 36 units squared, what is the length of a side to the nearest tenth?

I have been working with finding the area of a regular triangles, squares, and hexagons using special right triangle formulas drawn from the radii and apothems, but I cannot for the life of me work ...
8
votes
1answer
247 views

Series for envelope of triangle area bisectors

The lines which bisect the area of a triangle form an envelope as shown in this picture It is not difficult to show that the ratio of the area of the red deltoid to the area of the triangle is ...
0
votes
1answer
88 views

Show that for triangle ABC, with complex numbers for the coordinates, that we have the following equation

so I am doing an assignment on triangles and complex numbers, but I am stuck in the very first question. I am not asking for the solution, I would just like a hint or some ideas on what I need to look ...
8
votes
2answers
177 views

How to prove that $\frac{r}{R}+1=\cos A+\cos B+\cos C$?

How do we prove that for any triangle this holds: $$\frac{r}{R}+1=\cos A+\cos B+\cos C$$ I can use this beautiful identity to prove several geometric inequalities, but I have no idea how to prove the ...
0
votes
1answer
33 views

Reverse triangle inequalities with three elements

Could you help me to show that $$ |a-b-c|\geq |b|-|a|-|c| $$ ?
1
vote
1answer
89 views

Finding ratio of external division in a triangle.

Given a $\triangle ABC$ and $P$ dividing $AB$ internally in the ratio $2:3$ $Q$ dividing $AC$ internally in the ratio $1:2$ , with $PQ$ produced and $BC$ produced intersecting in $R$ , to find the ...
2
votes
2answers
297 views

triangle construction given side, angle and median

I can't figure out the solution to this, it looks to me like it doesn't have any solution but I need some proof. problem: Construct a triangle ABC with given $a=6 cm$ $\alpha=75^\circ $ and ...
3
votes
2answers
155 views

Given the length of two altitudes and one side , find the area of triangle.

Segments $BE$ and $CF$ are the altitudes in $\triangle ABC$. $E$ is on line $AC$ and $F$ is on line $AB$. $BC = 65$, $BE = 60$ and $CF = 56$. Find $A(\triangle ABC)/100$. By the Pythagorean ...
4
votes
3answers
90 views

In $\triangle ABC$, I is the incenter. Area of $\triangle IBC = 28$, area of $\triangle ICA= 30$ and area of $\triangle IAB = 26$. Find $AC^2 − AB^2$

In $\triangle ABC$, I is the incenter. Area of $\triangle IBC = 28$, area of $\triangle ICA = 30$ and area of $\triangle IAB = 26$. Find $AC^2 − AB^2$. Here is a sketch that I drew: From the given ...
0
votes
4answers
134 views

Midpoints of a triangle [closed]

The points $(4,2), (-1,-3)$, and $(-10,6)$ are the midpoints of the sides of triangle $ABC$. What is the area of triangle $ABC$?
0
votes
1answer
33 views

triangle with given 2 medians and 1 side

I need help with this exercise I got. We have a triangle with given medians ma=6, mb=9 and side (without given median on that side) c=6. What is the length of a and b and with what values of ma,mb ...
3
votes
2answers
221 views

A question related to triangles , areas , ratio of areas of triangles.

I know the title is confusing but that is because of 150-character limit, if anyone of you can improve it , please do. Consider $\triangle ABC.$ Choose a point $D$ on segment $BC$ such that ...
1
vote
0answers
29 views

How to prove that $FC/FA + GC/GA= 0$ from this triangle problem?

In triangle $ABC$, a transversal line intersects $AB$, $BC$, $CA$ at $D,E,F$ respectively. $BS$ intersects $AC$ at $G$, where $S$ is the intersection of $AE$ and $CD$. How to prove that ...
1
vote
2answers
94 views

A problem related to circle , altitude , triangle.

Consider a $\triangle ABC.$ Draw circle $S$ such that it touches side $AB$ at $A$. This circle passes through point $C$ and intersects segment $BC$ at $E.$ If Altitude $AD ...
2
votes
2answers
257 views

In △ABC, median AM = 17, altitude AD = 15 and the circum-radius R = 10. Find BC^2

Question is as per title. Here is a sketch that I made : By Pythagorean theorem , DM is 8. Now how can I calculate BD and MC? I still haven't found a way to utilize the information that the ...
0
votes
1answer
71 views

why a^2 + b^2 = c^2 in right-angled triangle [duplicate]

a^2 + b^2 = c^2 what is the demonstration of this rule with triangle which has 90 deg? can be proofed using geometry?
1
vote
0answers
83 views

How to prove these equations base on this following interior and exterior angle bisectors problem?

In the triangle $\triangle ABC$, length of $BC$ is larger than length of $AC$. The interior angle bisector of $\angle C$ intersects $AB$ at $D$; and the exterior angle bisector of $\angle C$ ...
0
votes
0answers
30 views

How to prove that PH is containing midpoint of side MN from this circle and triangle problem?

Given: triangle ABC is acute triangle. M and N are midpoints of AB and BC respectively, while BH is altitude of triangle ABC. Circles AHN and CHM meet at point P. (P is not same with H) How to ...
4
votes
2answers
147 views

In triangle, $\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$

To prove $$\sin\frac{A}{2}+\sin \frac{B}{2}+\sin\frac{C}{2} -1 = 4\sin \frac{\pi -A}{4}\sin\frac{\pi -B}{4} \sin\frac{\pi-C}{4}$$ My approach : $$ \begin{align} \text{L.H.S.} & = ...
3
votes
1answer
149 views

What is the converse of the triangle inequality?

It's usual when presenting a theorem to also present its converse. Surprisingly, I've never seen the triangle inequality's converse stated. Triangle inequality: If the sides of a triangle are a, b, ...
1
vote
1answer
40 views

How to prove that:$ BC^2= 3CM^2 + AC^2 $from this triangle problem?

In the triangle $\triangle ABC$, angle $\angle A$ is larger than angle $\angle B$. We choose points $M$ and $N$ at $AB$ such that $AM=MN=NB$. How to prove that: $BC^2= 3CM^2 + AC^2$? Which ...
0
votes
1answer
55 views

what is the measure of angle ECD from this following triangle problem?

In triangle ABC, AB is larger than BC. Then, we choose point E outside the triangle such that BE=BC. We extend line AB to D, such that BD=BC. BF is angle bisector of angle ABC. If DC is parallel ...
4
votes
2answers
66 views

How to prove that the angle between two sides of that triangle is less than 60 degree?

The product of two sides of triangle is equal to 8*(R*r) where R is circumradius of this triangle, and r is inradius of this triangle. How to prove that the angle between two sides of that triangle ...
0
votes
2answers
67 views

MCA entrance question

In triangle $ABC$, the value of $\ \displaystyle \sum_{r=0}^n\ ^nC_ra^rb^{n-r}\cos(rB-(n-r)A)$ is equal to (a) $c^n$ (b) $b^n$ (c) $a^n$ (d) $0$ I have no idea how to start ...
0
votes
0answers
37 views

I stumbled when i saw this (Travelling Salesman related)

I have here 5 locations like below I then have a tour of 1,2,3,4 (point 5 isn't inserted into tour yet) like below I then find the shortest addition distance to include point 5 into tour, the ...
5
votes
2answers
210 views

Minimum area of a triangle

In triangle inscribed circle with radius $r = 1$ and one of it sides $a=3$. Find the minimum area of triangle? Ans = 5.4 My reasonings: $BC = a$, $AC = b$, $AB = c$ $AD=AF=x$ $FC=CE=y$ ...
1
vote
2answers
108 views

Three circles with two common points

Let $ABC$ be a triangle of any type and $A_1,B_1,C_1$ the feet of the heights. Denote $M,N,P$ the orthogonal projections of the point $A$ onto the lines $B_1C_1,C_1A_1$ and $A_1B_1$. The circes ...
0
votes
3answers
69 views

Triangle inequality and its equality

How do I prove this? $$|x+y|=|x|+|y|\Leftrightarrow xy\geq0$$ I tried to use the triangle inequality, but I didn't get so far... Thanks!
0
votes
2answers
85 views

Why can I not use an equation using proportions to solve this triangle problem?

It is difficult to see the picture of the problem. The question is "What are the lengths of AC and AB?" What is given is a right triangle, ABC. Angle B is 30 degrees and BC is 7.0 distance. The ...
0
votes
1answer
143 views

Competition math geometry question

The perimeter of triangle ABC is $36$, and its area is $36$. Compute $\tan\frac{A}2 \tan\frac{B}2 \tan\frac{C}2$. I found that the answer is $1/9$, but I was not able to find a reason for this. Could ...
0
votes
1answer
60 views

How to prove that $DE=EF +DG$ from this following triangle problem?

Given a right triangle $ABC$, where $C$ is a right angle. We choose points $G$ at $AC$ and $F$ at $BC$, and $D$ and $E$ at $AB$. We draw right triangles $AGD$ and $EBF$, such that $\angle AGD= ...
2
votes
1answer
77 views

Solution for the value of an angle of a triangle ABC

Find value of angle m< DBC Where $$BD=DC=AC$$ $$2(m\langle BAC)=14(m\langle ABD)=7(m\langle BCD)$$ I tried hard but im out of ideas now, I know the answer is 20 but I want to know how, thanks ...
25
votes
4answers
1k views

Two circles inside a right angled triangle!

The other day I was playing with Ms Paint drawing circles here and there - I coincidentally drew a circle inside a right angled triangle which I already drew. Strangely A problem struck to my mind ...
2
votes
1answer
24 views

tetrahedron height

I've got the next figure: Now I would like to calculate the height, so from D to the plane ABC. First, I've tried with a coordinate system, but it's to difficult to take these distances into ...
1
vote
0answers
75 views

solve this complex triangle question ?

,D,E,F are midpoint of triangle ABC on sides BC , CA , AB. The feet of the altitudes from A,B,and C are P,Q and R. h is the orthocentre and O is the circumcentre . Then prove 2OD=AH. The nine ...
0
votes
3answers
129 views

Finding area of sector inside an triangle

I have been asked this question from a junior and could not solve the question in a simple way. I am asking help on this platform. For a triangle $ABC$, Points $D, E$ on $AB$, where ...
0
votes
1answer
40 views

Calculate regular or equilateral triangle altitude with radius only possible?

I need to calculate the altitude of a regular triangle (equilateral) but i only have the radius (polygon radius) available (http://www.mathopenref.com/polygonradius.html). I have been searching for ...
0
votes
2answers
90 views

To find base and height of an isosceles trangle if sides and area are give

The area of an isosceles triangle is $60cm^2$ and the length of equal side is $13cm$. Find height and base.
4
votes
1answer
132 views

Length of median extended to the circumcircle

A triangle has side length $13,14,15$, and its circumcircle is constructed. The median is then drawn with its base having a length of $14$, and is extended to the circle. Find its length.
2
votes
0answers
97 views

Hijacked Malaysian plane position geometry

Sorry to get geeky in the midst of a tragedy and likely horrible crime, but does anyone know how they got this diagram showing the possible last known positions of the possibly hijacked Malaysian ...
0
votes
1answer
377 views

Maximum perimeter of an isosceles triangle inscribed in the unit circle?

So I have seen this question asked before but with variations (circle of radius 4, and an equilateral triangle) and so I am hoping for an answer on how to do this. After looking around I saw that ...
1
vote
1answer
33 views

Solving integral including a triangle

How can I solve this integral? Image link: http://oi61.tinypic.com/2jeoga1.jpg I tried to solve it: x^2/2 from 4 to 0. [(4^2/2)-(0^2/2)]=8 but its wrong. Do I have to multiply base*height/2 because ...
-1
votes
5answers
174 views

Trigonometry Question (finding the sin, cos, cosec etc on a right-angled triangle)

For the right-angled triangle $\widehat{PQR}$, where $\overline{PQ} = 9\text{ cm}$, $\overline{QR} = 40\text{ cm}$ and $\overline{PR} = 41\text{ cm}$, give the value of: a) $\sin \hat{P}$ b) $\cos ...
3
votes
2answers
184 views

Find the maximum angle possible

$P$ is a point on the $Y-axis$ . Find the maximum possible value of $\angle APB$ where $A=(1,0)$ and $B=(3,0)$. Here is how I solved the problem. Suppose $P=(0,k)$ . Then using the cosine formula we ...
0
votes
1answer
36 views

Calculate an angle between time 00:00 and a mouse cursor position

I have to build a timepicker where user clicks on a clock like circle and it gives a time. Once I have cursor position I think that all I have to do is to calculate an angle between time 00:00 and a ...
1
vote
1answer
45 views

Similar Triangles with proportions

In $\triangle ABC$, $AB=8, BC=7, CA=6$, and side $BC$ is extended to point $P$, so that $\triangle PAB$ is similar to $\triangle PCA$. Find the length of $PC$.
0
votes
2answers
47 views

Prove that this is an isoceles triangle

I'm trying to solve a problem here. It says: "Prove that a triangle is isoceles if $\large b=2a\sin\left(\frac{\beta}{2}\right)$." $B-\beta$ I've tried to prove it but I can't Can anyone help me?
0
votes
1answer
31 views

Representation of a Triangle

In this document, a Triangle is represented as: $$ T(s,t) = B + sE_0 + tE_1\\for~all~(s,t)\in D=\{(s,t):s\in[0,1], t\in[0,1],s+t\le1\} $$ Can someone explain this representation of a Triangle?