For questions about properties and applications of triangles

learn more… | top users | synonyms

0
votes
2answers
24 views

Prove that DE || BC

Let M be the midpoint of side BC in triangle ABC. The angle bisector of BMA intersects AB in D, while the angle bisector of CMA intersects AC in E. How can i prove that DE||BC? I drew out the ...
2
votes
2answers
38 views

Is there a way of determinine the side lengths of a isosceles triangle knowing its angles and area?

I want to be able to determine the side lengths (or at least one side length) of an isosceles triangle knowing only its surface area and angles. Is this possible?
-1
votes
1answer
38 views

How many different shapes can you construct with n equilateral triangles?

If you have n equilateral triangles, and you want to connect them all to each other at the edges, how many different shapes can you make? Triangles are identical in size and shapes that are ...
2
votes
5answers
68 views

what is the value of angle A

The triangle ABC is random. The line $AD$ is twice big as the line $DC$ ($AD=2*DC$). We know only the two angles that are shown in the picture. What's the value of angle $A$?
1
vote
1answer
28 views

Median BM of triangle ABC two results

Given Calculate the measure of the median $\overline{BM}$ of ABC triangle, given A (-6.1); B (-5,7) and C (2,5) I get this result: $Xm = \frac{Xc - Xa}{2} + Xa$ $Xm = \frac{2-(-6)}{2} + (-6) = ...
19
votes
11answers
5k views

In a right triangle, can $a+b=c?$

I understand that due to the Pythagorean Theorem, $a^2+b^2=c^2$, given that $a$ and $b$ are legs of a right triangle and $c$ is the hypotenuse of the same right triangle. However, most of the time, ...
2
votes
0answers
51 views

Prove that the intersection point of lines $AK$ and $CL$ lies on the line $BO$

$AA', BB'$ and $CC'$ heights of an acute triangle $ABC$. The circle with center $B$ and radius $BB'$ intersects the line $A'C'$ in the points $K$ and $L$. Prove that the intersection point of lines ...
2
votes
1answer
18 views

All triangles that have the same orthocenter and circumcircle have the same nine-point circle

True or false? Prove it. I guess it would help to figure out whether 2 triangles can have the same circumcenter or orthocenter and not be congruent. I have no clue how to figure this out. If they ...
0
votes
0answers
12 views

A new family circle associated with the Tucker hexagon and the Symmedian point

I am looking for the problem following: Let ABC be a triangle, let $A_1B_1C_1$ be a cevian triangle of the symmedian point. Let $B_aC_aC_bA_bA_cB_c$ be a Tucler hexagon of $ABC$. Such that $A_bA_c ...
0
votes
0answers
12 views

A generalization of the first Droz-Frany circle

I am looking for a proof of the following problem: Let $ABC$ be a triangle with circumcenter $O$, and the medial triangle $M_aM_bM_c$. Let $O_a, O_b, O_c$ be three points on three lines $OA, OB, ...
2
votes
1answer
31 views

What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles?

What triangles can be cut into three triangles with equal radii of the circumscribed circles around these triangles? My work so far: Case 1) let $ABC -$ an acute-angled triangle. Then radii of the ...
1
vote
1answer
38 views

How to solve this problem about the reciprocals of the angles inside an acute triangle?

I was solving some previous year papers and stumbled upon this question. I don't even know how to approach this problem. The problem is as follows: Consider an acute angled triangle $PQR$ such ...
0
votes
1answer
18 views

Ratio of areas in triangles with Cevians

Can someone show me how to solve this problem as they normally would? It would also be very helpful if you could also give me a mass points solution. I know mass points works for ratios of segments, ...
0
votes
2answers
32 views

Geometry and property of triangle

$ABC$ is a cyclic triangle and bisector of angle $B\widehat{A}C$, $A\widehat{B}C$ and $A\widehat{C}B$ touches circle at $P$, $Q$ and $R$ respectively then measure of angle $R\widehat{Q}P$ is? The ...
1
vote
3answers
52 views

Cover a polygon with polygons

Besides right angled triangles, is there any polygon I could use to cover any given (regular or not) polygon? It's clear that given a triangle, square, hexagon or rectangle you would other options. ...
0
votes
1answer
28 views

How to do Vectors with triangles?

This is my homework and I have just started learning it and I don't really quite understand it.
5
votes
4answers
131 views

In $\triangle ABC$, if $\cos A\cos B\cos C=\frac{1}{3}$, then $\tan A\tan B+\tan B \tan C+\tan C\tan A =\text{???}$

In $\triangle ABC$, if $$\cos A \cos B \cos C=\frac{1}{3}$$ then can we find value of $$\tan A\tan B+\tan B \tan C+\tan C\tan A$$ ? Please give some hint. I am not sure if $\tan A \tan ...
1
vote
2answers
34 views

Prove that an Equilateral cannot have natural number points

Let $ OAB $ be an equilateral triangle with $O(0, 0),\ A(m, n),\ B(x, y)$, where $m, n \in \mathbb{N}^{\ast}$ and $x, y \in \mathbb{R}_{+}$. Prove that $B$'s coordinates can't be both natural ...
2
votes
1answer
27 views

General Triangles: Area, lengths and angles calculations

I have a question on General Triangles (as in non right angle). I’m trying to create a program that calculates angles and sides based on the user entering Area and some sides length or angle ...
0
votes
1answer
62 views

Given latitude and longitude, how to find central angle and arc length of spherical triangle?

Lewis and Clark followed several rivers in their trek from what is now Great Falls, Montana, to the Pacific coast. First, they went down the Missouri and Jefferson rivers from Great Falls to Lemhi, ...
2
votes
0answers
27 views

Find the length of a triangle

Question is: Find the length of $\text{AO}$ and $\text{BO}$ My work, with the things I know already: Length: $\text{CO}=r$ and $\text{TO}=\frac{r}{4}$ and $\text{TG}=\frac{r}{2}$ and ...
0
votes
0answers
8 views

Sampling from Irwin-Hall distribution using triangular distribution

So I need to sample from the Irwin-Hall distribution using rejection sampling with the triangular distribution. I built 2 functions: The first is d_irwin which receives an $x\in supp(g)$ and the n we ...
0
votes
0answers
64 views

Calculate the coordinates of two points in an isosceles triangle

In the triangle below, given the point A, angle θ and length d of the two equal sides, how can the points B and C be calculated? Edit:After brainstorming for quite some time, I ended up with a ...
3
votes
1answer
88 views

Relationship between incenter and circumcenter

Let ABC be an acute triangle with circumcenter O and incenter I. Points E, M lie on AC and F, N on AB so that BE ⊥ AC, CF ⊥ AB, ∠ABM = ∠CBM and ∠ACN = ∠BCN. Prove that I lies on EF if and only if O ...
0
votes
3answers
44 views

Find area of triangle ABC given areas of sub-triangles

The line p is parallel to the the side AB of triangle ABC and splits the sides AC and BC in points D and E, respectively. If the area of triangle ABD is m and the area of triangle AEC is n, find the ...
1
vote
2answers
41 views

Finding sides of triangle

Given : $$\triangle ABC$$ $$M \in AB,N \in BC ,P \in AC$$ are the points at which the incircle crosses the triangle $$MN=3\sqrt{10}$$ $$NP=2\sqrt{20}$$ $$PM=10$$ I have to find the sides of the ...
1
vote
0answers
44 views

Taylor's formula and its quadratic term

I struggle with the following problem: For a function $$f: \mathbb{C} \rightarrow \mathbb{R}~,$$ $f$ attains its maximum for $z_0= e^{i\pi/3}$, $f(z_0)=F_{max}.$ Assume we may use Taylor's theorem ...
3
votes
1answer
36 views

Circumcentre and Incentre [closed]

if I is the incentre and S is the circumcenter of ABC prove that angle IAS is half the difference between angle B and angle C
2
votes
2answers
77 views

Circle inscribed in Equilateral Triangles

The circle inscribed in the triangle $ABC$ touches the sides $BC$ , $CA$ , and $AB$ in the points $A_1,B_1,C_1$ respectively. Similarly the circle inscribed in the triangle $A_1B_1C_1$ touches the ...
1
vote
0answers
181 views

Inequalities of the triangle

I created a inequality below, what you think guys? Let $ \alpha',\beta',\gamma'$ be angles of an acute triangle, and let $ n\in\mathbb{N}^{*}$ and $\displaystyle j\in\mathbb{N}$. Prove that: $$ ...
2
votes
1answer
45 views

Point which minimizes the squared sum distances to edges of a triangle [duplicate]

I need to find coordinates of a point at which the sum of the squared distances to each of the three lines that form a triangle is minimized. It seemed to me that the point is the triangle's ...
0
votes
1answer
23 views

X is any point on AB and the median AD of triangle ABC meets XC at Y.Prove that XY/YC = AX/XB

X is any point on AB and the median AD of triangle ABC meets XC at Y.Prove that XY/YC = AX/XB.
0
votes
1answer
31 views

Write down the iterated integral which expresses the surface area of z = y 6 cos3 x over the triangle with vertices (-1,1), (1,1), (0,2):

This problem has been done a few times on other sites but there is no work or explanation of the steps taken to get h(x,y). I can understand getting the limits by finding the bounds of the triangle ...
0
votes
1answer
40 views

Finding angle associated with point inside an equilateral triangle.

$\triangle{ABC}$ is an equilateral triangle. $|AD|=6$. $|BD|=10$. $|CD|=8$. What is $m\angle{CDA}$? First thing comes to mind is Ceva theorem. I used its trigonometric form to reach ...
2
votes
1answer
24 views

triangle park problem [closed]

We have a park that is triangle. We don't know the shape of the triangle and it can have any triangle shape and lengths. Where should we place a lamp to have light everywhere in the park? my english ...
0
votes
1answer
24 views

Pythagorean Theorem on Spiral of Theodorus Triangles

I have 1 right triangle of dimensions $\sqrt75$$, 11, 14$. I'd like to know how to quickly obtain the other right triangles with $\sqrt75$ as a leg, and two integers as the hypotenuse and the other ...
0
votes
1answer
26 views

Get a point by a given point, degree and distance

I have a point $(x_1, y_1)$, an angle $A$ and a distance $D$ How do I get the point $(x, y)$ which is $D$ unit away from $x_1, y_1$ with angle $A$. In the right image, the answer will be $(x_1 - ...
0
votes
2answers
40 views

geometry - prove that you can make new triangle with..

I have a triangle, the length of heights are $i,h,g$. Prove that we can build a new triangle so that the lengths of the sides are: $i^{-1}, g^{-1}, h^{-1}$ (see picture)
2
votes
2answers
66 views

Joint PDF of two random variables in a triangle

Let the random variables $X$ and $Y$ have a joint PDF which is uniform over the triangle with vertices at $(0, 0), (0, 1 )$ and $(1, 0)$. Find the joint PDF of $X$ and $Y$. So ...
5
votes
2answers
1k views

Determine the third point in right triangle only knowing the coordinates of the other two points

I have a right triangle $ABC$. I am given the coordinates of the two points $A(x_1, y_1)$ and $C(x_2, y_2)$. Given points $A$ and $C$, I want to determine the coordinates of $B$. I know there are two ...
0
votes
0answers
30 views

Area of non-spherical triangle on a sphere

This is a followup to the question Area of triangle on a sphere (not spherical triangle) Since it's now almost two years later, I'm making it a new question. The problem is to find the area of ...
-1
votes
1answer
42 views

Rotate right triangle with perimeter 1 about the hypotenuse [closed]

We rotate every right triangle with perimeter 1 about its hypotenuses. Is it true that we can choose a solid from so obtained solids that has maximum volume? If yes, what's the volume? I guess I ...
0
votes
1answer
35 views

The ratio of areas of two triangles with the same altitude is equal to the ratio of their bases

How can we prove that the ratio of areas of two triangles of equal altitudes is equal to the ratio of their bases?
2
votes
1answer
90 views

Inequality involving circumradii

Let $ABC$ be a triangle and $M$ a point on the side $BC$. Let $R_1$,$R_2$, and $R$ be the circumradii of the triangles $ABM, ACM$, and $ABC$. Show that $\max\{R_1,R_2\} \geq R\cos\frac A 2$.
2
votes
2answers
66 views

Three Altitudes of a triangle are concurrent

I have been told that this well known fact can be shown using only Euclid's propositions from books one to three, and cyclic quadrilaterals. I can't figure out how to start, which quadrilateral ...
2
votes
1answer
28 views

Law of Cosine formula that I can't seem to rearrange.

I was in midst of solving a trig problem, and it required using the formula of Law of Cosine. For my case, I had to solve for a specific variable which was $\cos (A)$. Would you show me step-by-step ...
1
vote
1answer
37 views

Equilateral triangle with vertices whose coordinates on the Cartesian plane are integers. Does such a triangle exist? [duplicate]

Can you build an equilateral triangle on a Cartesian plane whose vertices only have integer values as their coordinates? Looking at the simplest example, i.e. a triangle with vertices (0,0), (1,0) ...
0
votes
1answer
33 views

Geometry: Determining the length of a side of a triangle [closed]

Triangle $ABC$ has all sides of integral length. Vertex $A$ is at $(0,0)$, $B$ lies on the line joining $(0,0)$ and $(3,6)$ and $C$ lies on the line joining $(0,0)$ and $(2,-1)$. Two of the three ...
1
vote
1answer
32 views

Finding the length of a side of an equilateral triangle

There is a large right isosceles triangle with a hypotenuse length of $24$. Inside the triangle is an equilateral triangle with a vertex on the midpoint of the hypotenuse. If the length of each side ...
0
votes
1answer
31 views

Solving for length of an unknown side of a triangle.

I have been given the figure below: Figure (click me). I know that $AD=20-x$ and $m\angle ACD=m\angle BCD$. How can I set up a ratio also knowing that $AC=11$ and $BC=14$ in order to find $x$? ...