For questions about triangles

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19
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1answer
676 views

Are there prime lengths in triangle with all integer sides and heights?

Suppose you have a triangle in which all sides and all heights are integer in length (i.e. triangle with sides 20, 25, 15 has heights 15, 12 and 20). Could it be that at least one of those numbers is ...
0
votes
1answer
59 views

is there a formula for working out the angles of a triangle to make the sides meet at the top?

I am doing a GCSE maths foundation paper for revision and one question has a triangle with the base side being 9cm and the other 2 sides 7.5cm. Is there a formula for finding the angles of the ...
1
vote
2answers
92 views

Minimizing area of a triangle with two fixed point and a point on parabola

A triangle is made up of three points, $A, B$, and $P$. $A(-1, 0)$ $B(0, 1)$ $P$ is a point on $y^2 = x$ Minimize the area of Triangle $ABP$. My approach is far too complicated, which ...
7
votes
1answer
795 views

Sum of distances from triangle vertices to interior point is less than perimeter?

Let $M$ be a point in the interior of triangle $ABC$ in the plane. Prove $AM+BM+CM<AB+BC+CA$. The above question was posed to someone I know who is taking high-school Euclidean geometry. I'm ...
4
votes
4answers
4k views

Find the legs of isosceles triangle, given only the base

Is it possible to find the legs of isosceles triangle, given only the base length? I think that the info is insufficient. Am I right?
2
votes
1answer
57 views

Find the value of the expression $\frac{AF}{FD}$.

In the given figure $AC=BD=3$ units and $CE=DE=1$ unit. Find the value of the expression $\frac{AF}{FD}$. We know that the median divides the area of a triangle in to two equal halves. Therefore, ...
0
votes
2answers
153 views

Sides of the Right angled Triangle in Complex notation.

If $z=a+ib$ is a complex number, then $z, iz, z+iz$ represents sides of the right angled triangle. I got this result through Cartesian form, i,e. $(a,b),(-b,a) and (a-b,a+b)$ are the vertices of the ...
3
votes
1answer
263 views

Algorithm to find rectangle inside a triangle

I am trying to write a program that generate procedural cities. However, I am stuck on a problem : I don't know how to subdivide a triangle into a rectangle and other triangles. I know how to ...
5
votes
2answers
441 views

When is the area of a triangle whose side lengths are consecutive integers also an integer?

Consider a triangle with side lengths 3, 4, and 5. By Heron's formula, its area is $\sqrt{6(6 - 5)(6-4)(6 - 3)} = \sqrt{6(1)(2)(3)} = \sqrt{36} = 6$. Are there any other triangles like this?
0
votes
1answer
148 views

Drawing a triangle in a unit circle

This is a question that I derived for a long time ago. It asks if we draw a triangle in a unit circle does all arc lengths $(\alpha ,\beta ,\theta)$ and sides of triangle $(a,b,c)$ can be rational ...
1
vote
1answer
114 views

Need help solving -

i was writing an paper on solutions of triangles when i encountered this sum - In a $\Delta$ ABC , P is an interior point such that $\angle PAB = 10^\circ$ , $\angle PBA = 20^\circ$ , $\angle PCA = ...
0
votes
3answers
103 views

Similar triangles question

If I have a right triangle with sides $a$.$b$, and $c$ with $a$ being the hypotenuse and another right triangle with sides $d$, $e$, and $f$ with $d$ being the hypotenuse and $d$ has a length $x$ ...
2
votes
5answers
144 views

How can I solve this Laws of Sines problem?

This is a homework question that was set by my teacher, but it's to see the topic our class should go over in revision, etc. I have calculated $AB$ to be 5.26m for part (a). I simply used the law ...
2
votes
1answer
106 views

Property of bisectors of right triangle

In triangle $ABC$ $\angle C=90^\circ$, $AA'$ and $BB'$ are angle bisectors intersecting at $I$ ($A'\in BC$, $B'\in AC$). What would be the easiest way to prove that projection of $I$ onto $AB$ lies in ...
6
votes
4answers
1k views

How to find area of triangle from its medians

The length of three medians of a triangle are $9$,$12$ and $15$cm.The area (in sq. cm) of the triangle is a) $48$ b) $144$ c) $24$ d) $72$ I don't want whole solution just give me the hint how ...
0
votes
1answer
59 views

find area of Triangle ABF

In the figure given below, rectangle $CDEF$ with perimeter $32$ has the maximum area. Find the area of the triangle $ABF$ So, I tried the following $P = 2W+2H$ where $P$ is given $32$. I am not ...
-1
votes
2answers
123 views

Find area of triangle ABC

BD Perpendicular AC , AB =BC=a Find the area of triangle ABC I have tried Googling , I used formula 1/2 (base X Height) . Used Pythagorean theorem. Anyone can suggest me solution.
2
votes
3answers
350 views

Geometry - Equilateral triangle covered with five circles

I have to cover an equilateral triangle (whose sides are 1m long) with 5 identical circles: what's the minimum radius of the circles?
2
votes
4answers
222 views

Length of Triangle BCD

Hey, well I'm doing some higher level revision and I'm stuck... In the diagram triangle BCD is mathematically similar to triangle ACE. So what is the length of BD? How do you work it out?
1
vote
1answer
48 views

point on triangle

let we have a domain, for example square that partitioned by some triangles that are not necessarily similar, and we know the coordinates of all vertices and the maximum size of the sides of all ...
1
vote
2answers
227 views

Can all possible angles on a rational triangle be represented as a rational multiplied by Pi?

The reason I ask: I was wondering if it was possible to find the angle of a rational triangle by only using the lengths of its sides and knowledge of $\pi$ (that is, no inverse trig functions). So, ...
0
votes
1answer
39 views

a question about triangle

Let $(x_1,y_1),(x_2,y_2),(x_2,y_2)$ are the vertices of the triangle T. I want to show that the line $L(\alpha_3)$ defined by $$x=(1-\alpha_3-\alpha)x_1+\alpha x_2+\alpha_3 x_3$$ ...
1
vote
0answers
106 views

Proving that the circumcenter is the centroid

Given a triangle and its centroid, we know that the 3 line segments between the centroid and each of the vertices of the triangle divide the triangle into three smaller triangles. Prove that the ...
0
votes
1answer
57 views

To find a point in a horizontal plane to minimise the distance

A, B and C are three points in 3D space. The points A and B are fixed. C is below A and B. $H_{p}$ is any given plane. Point C moves along the plane $H_{p}$. How to find the location of point C ...
1
vote
0answers
82 views

maximum length of a scaled vector in a triangle (simplex)

Given a triangle (or, in general, a simplex) $T$ and a vector $\vec{s}$, I'd like to compute the quantity $$ \max\{|x-y|: x,y\in T, x-y = \alpha \vec{s}, \alpha\in\mathbb{R}\} $$ i.e., the maximum ...
1
vote
1answer
118 views

Triangle optimization problem

Let $a,b,c$ be the sides of a triangle , then what is the maximum and minimum values (if exist) of the following quantities (i) $\dfrac {a^2b^2c^2}{(a+2b)(a+2c)(b+2c)(b+2a)(c+2a)(c+2b)}$ (ii) ...
0
votes
3answers
2k views

Calculating circle radius from two points and arc length

For a simulation I want to convert between different kind of set point profiles with one being set points based on steering angles and one being based on circle radius. I have 2 way points the ...
2
votes
1answer
72 views

Angle sum of a triangle.

Can you please describe the geometry in which the sum of the angles of the triangle can be less than 180 degrees?
0
votes
2answers
20 views

Triangle inequlity improvment with the angle conditions

I was working on how to proof $a+b \leq x+y+z$? Apply triangle inequity to the triangle ADC, $x+z \geq a$ Apply triangle inequity to the triangle DCB, $y+b \geq z$ Adding above inequities, ...
1
vote
1answer
75 views

P is a point in triangle $ABC$, what is $[APC]$?

Moderator Note: This question is part of an ongoing contest on Brilliant.org, and will be unlocked in 1 week. P is a point in triangle $ABC$. The lines $AP$,$BP$, and $CP$ intersect the sides ...
1
vote
1answer
97 views

Similarity of triangles in a circle

The problem: c is a circle with a diameter AB. t is the tangent at the point B. Now C and D are two points on t and at different sides of B. I draw the line segments AC and AD, the point where AC ...
3
votes
4answers
208 views

Maximum surface inside a triangle

If I have a triangle with sides of length a, b, c and I have a rope of length L, what is the maximum surface of a boundary I can form with that rope that is entirely inside the triangle. Normally, ...
2
votes
2answers
629 views

Getting an acute angle for an obtuse angle using law of Sines.

I have done this problem over and over again. I even looked up tutorials on how to properly use law of sines. It's rather embarrassing that I'm struggling so much wish this simple trigonometric stuff. ...
4
votes
4answers
346 views

Is every prime number the leg of exactly one right triangle with integer sides? What's wrong with my argument that this is impossible?

The problem is: "prove that every prime number is the leg of exactly one right triangle with integer sides." However, I seem to have proved that this is impossible. What did I do wrong here? Let ...
2
votes
2answers
114 views

Congruent Triangles

Triangle ABC and Triangle DBC are two isosceles triangles on the same base BC and vertices A and D are on the same side of BC.If AD is extended to intersect BC at P,show that (a)Triangle ABD ≈ ...
5
votes
1answer
228 views

Sum of medians of a triangle

I'm very confused because I don't know how I can prove that the sum of the medians of a triangle is equal to the vector zero. Can someone give me a tip or something? Thanks! (And sorry if this ...
0
votes
1answer
220 views

Gergonne Point of a triangle coinciding with other triangle centers

I am trying to prove the following: Let $T$ be the Gergonne point (the intersection of the lines that connect the points of tangency of the incircle with the vertices of the triangle) of $\triangle ...
2
votes
1answer
81 views

Is there an exploitable relationship between the angles of AB and AC in a triangle when moving point A?

I'm writing software to dynamically create a random, generated world (civ-esque), populate it with settlements, and draw roads between them. It would be relatively easy to draw roads like this: ...
4
votes
2answers
150 views

Coordinates of parallel triangle with a distance of 'd' between the parallel edges?

I have a triangle with Co-ordinates $\{(x_1,y_1),(x_2,y_2),(x_3,y_3)\}$. I need to find co-ordinates of a triangle,whose edges are exactly $\alpha$ distance from previous triangle. Below is the figure ...
1
vote
4answers
205 views

Triangle proof using law of sines

In triangle $ABC$, suppose that angle $C$ is twice angle $A$. Use the law of sines to show that $ab= c^2 - a^2$.
4
votes
2answers
127 views

Minimizing the length of a pipeline between cities

I have been trying to minimize piping going to two different cities. City A is located at $(0,4)$ and city B is located at $(6,3)$. The cities must connect to the $x$-axis (the main pipe line.) It ...
1
vote
1answer
620 views

arithmetic progression of triangle sides

Let $gcd(a,b,c)=1$ such that $a^2, b^2, c^2$ are in arithmetic progression. Show they can be written in the form $a=-p^2+2pq+q^2$ $b=p^2+q^2$ $c=p^2+2pq-q^2$ for relatively prime integers $p,q$ of ...
1
vote
2answers
105 views

How to find co ordinates of a triangle after increasing the area by a factor of $\alpha$?

i am given with a triangle $\{(x_1,y_1),(x_2,y_2),(x_3,y_3)\}$ and the area need to be increased by a factor $\alpha$. can i anyone let me know formula to find the co ordinates of new triangle? There ...
0
votes
1answer
106 views

Triangle $z$-index interpolation between the vertices

I got a $2$D triangle, each vertex has a $2$D coordinate with a $z$-index value (NOT a $z$ coordinate!). The $z$-index value indicates whether a vertex lays on, in front of, or behind your screen ...
3
votes
1answer
311 views

Geometry - optimal 2D mesh between X expendable points

Say you have X points on a plane. If you connect two points, you form a line. Connecting three points forms a triangle. A line cannot cross a line, and a smaller triangle cannot be created inside a ...
1
vote
2answers
220 views

A question on Trigonometry (bisector)

If two bisector of a triangular is equal, then it is Isosceles triangular.
1
vote
2answers
145 views

Triangle $\Delta ABC$ , $a,b,c$ are in G.P.

If in a triangle $\Delta ABC$ the sides $a,b,c $ are in Geometric Progression.Find out the range of common ratio of the Geometric Progression. I understood that the twist is that we are bound under ...
0
votes
1answer
98 views

Law sines in Spherical Triangle $\rightarrow$ Law sines in plane triangle

Could any one tell me how to estimate or get law of sines in Spherical Triangle to The Law of Sines in Plane Triangle? i.e $\frac{\sin a}{\sin A}=\frac{sin b}{\sin B}=\frac{\sin c}{\sin C}$ to ...
2
votes
1answer
353 views

Finding side and angle of isosceles triangle inside two circles

I'm having a problem that I'm not sure how to solve (or if it's even possible). It's not homework, just something I'm struggling with for a project. :) Basically, there are two circles, represented ...
6
votes
2answers
138 views

Concurrency of A'L, B'M, C'N.

Need some help with the following problem. Problem: In $\triangle ABC$ the midpoints of $BC$, $AC$, $AB$ are $L, M,$ and $N$ respectively, and the points on the circumcircle opposite to $A, B,$ and ...