For questions about properties and applications of triangles

learn more… | top users | synonyms

0
votes
0answers
10 views

Find circcle radius by given triangle inside

So the triangle inside the circle: AB = 9cm CB = 6cm CH = 5cm I think solving this problem involves similar triangles. Thanks in advance, I'd like to have a ...
0
votes
0answers
9 views

How prove $S_{ABC}S_{XYZ}\ge S_{MNP}^2$ for an acute-angled triangle and $M, N, P$ are points from the segments $AB, BC, CA$ respectively

Let $ABC$ is an acute-angled triangle and $M, N, P$ are points from the segments $AB, BC, CA$ respectively. Let $CM\cup NP=X, AN\cup MP=Y, BP\cup NM=Z$. How prove $S_{ABC}S_{XYZ}\ge S_{MNP}^2$? ...
0
votes
1answer
14 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 ...
1
vote
1answer
22 views

Finding coordinates of the third point of a triangle from given?

In ABC triangle we know the coordinates of A and B vertices. We also know lengths of 2 edges shown in the picture and the third edge is calculatable. What is the most efficient functon to find x3 and ...
0
votes
0answers
12 views

What is the isotomic conjugate version of the six point circle of isogonal conjugates?

As it is well known, the pedal triangles of a pair of isogonal conjugates in a triangle share a circumcircle. This is a nice theorem, but is there an analogous version of it for a pair of isotomic ...
1
vote
2answers
43 views

Find distance between two poles.

2 poles, AB of length 2 metres and CD of length 20 metres are erected vertically with bases at B and D. The two poles are at a distance not less than twenty metres. It is observed that tan(angle(ACB)) ...
0
votes
0answers
30 views

Congruent Angles with Condition [on hold]

Let A be a point in the interior of triangle BCD such that $AB · CD = AD · BC$. Point P is the reflection of point A with respect to BD. Prove that $\angle PCB = \angle ACD$. I don't know how to ...
2
votes
3answers
89 views

Proving $ \frac{1}{c} = \frac{1}{a} + \frac{1}{b}$ in a geometric context

Prove or disprove $$ \frac{1}{c} = \frac{1}{a} + \frac{1}{b}. $$ I have no idea where to start, but it must be a simple proof. Trivia. This fact was used for determination of resistance of two ...
4
votes
4answers
255 views

In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. Why?

In a 30-60 right triangle the side opposite the 30 degree angle is half the length of the hypotenuse. A statement from the trigonometry section of Simmons' Precalculus in a nutshell. Please ...
0
votes
1answer
16 views

Calculate isocele triangle dimensions from angles [on hold]

maybe simple but I'm wondering how to dertermine dimensions of an isocele triangle from its given angles and a given height value. Any idea ? Thank you EDIT : this is what I may do : Lets say my ...
0
votes
1answer
9 views

How to get a Right Triangle's points' coordination in the space?

I have a Right Triangle with equal legs of 1 unit long rotated on 3 individual angles in the space like in the picture below: As could be seen in the picture, the input I have are the angles 'a' ...
0
votes
2answers
34 views

Prove the centroid coordinate formula

How to proof that the coordinate of the centroid of a triangle ABC is given by $\frac{A+B+C}{3}$ using vectors?
0
votes
0answers
67 views
+50

Cabri 3D - Rotating a triangle

I'm given the exercise, in Cabri 3D, to rotate the triangle T around the axis AB and lead it via the triangle To to the triangle T'. I tried to rotate the triangle T around a fixed point and then ...
0
votes
2answers
25 views

Layer on which ball belongs in tetrahedron

What is the most computationally efficient way to find the layer on which a ball (i) belongs when arranged in a tetrahedron or 3 dimensional triangle with a triangular base. The ball on the top layer ...
0
votes
0answers
9 views

trigonometry - find coordinates of inner triangle after rotation

here is my situation: I have a rectangle I'm rotating 30 degrees counterclockwise, how could I use trig to get the 3 vertices (corners) and lengths of the purple triangle sides and hypotenuse ...
5
votes
3answers
209 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
62 views

Find circumcenter when distance between ABC points of triangle with two points's ratio given

The complete problem is: I am having three points A,B,C whose ratio of the distances from points (1,0) and (-1,0) is 1:3 each. Then I need the coordinates of the circumcenter of the triangle formed ...
4
votes
2answers
57 views

How do I find the Intersection of two 3D triangles?

I've got a rather complicated geometry problem that I'm trying to solve - how to find the intersection between two triangles in 3D space. I've looked around at other questions and answers on this site ...
0
votes
1answer
19 views

Coordinate-geometry curiosity question

How can we draw a triangle give one of its vertex and the orthocentre and circumcentre? I tried to invoke the concept of 9 point circle and tried using the centroid but could not succeed in making ...
1
vote
1answer
26 views

Analytic-geometry rotation concept

I am confused how my book comes up with the following formula- Lets consider a Right angled Isoceles triangle with $2$ vertices on hypotenuse given as $(x_1,y_1)$ and $(x_2,y_2)$ Now the 3rd ...
0
votes
0answers
25 views

Get Tangential Vector from angular velocity

Good day, I'd like to know how to get the tangential vector with magnitude and direction (resolved through x y components) of a vector, given it's angular velocity. In this case, there is only 1 ...
0
votes
1answer
39 views

Orthocentre of a triangle

How do we determine the orthocentre of a triangle when the vertices are given as $(0,0),(x_1,y_1),(x_2,y_2)$? In a normal case i would take out the equation of any two perpendicular bisectors, get ...
8
votes
2answers
119 views

Eritrea's Theorem

According to this newspaper, an Eritrean high school student named Saied Mohammed Ali has discovered a new geometric theorem. Another source seems to say that it's the following: Say you have a ...
0
votes
2answers
49 views

Sum of the area of infinite similar equilateral triangles

How would I solve for the side depicted in the picture?
1
vote
2answers
117 views

Triangle containing most points from a set

Given a point set in $\mathbb{R}^2$, I need to find a triangle connecting three points of the set that contains the most points of the set. Points that lie on the connecting lines don't count. The ...
10
votes
5answers
2k views

Can area be irrational?

I'm stuck in a question of my book which says: If in an equilateral triangle the coordinates of two vertices are integral then what can we say about the coordinates of the third? The answer is that ...
0
votes
0answers
26 views

Angle condition for $a^2+c^2=nb^2$

Find a necessary and sufficient angle condition (independent of $a,b,c$ -- see under "what I have got so far" for examples) such that $a^2+c^2=nb^2$ where $n$ is a positive integer. Note: As usual ...
3
votes
2answers
66 views

In triangle $ABC$, $a^2+c^2=3b^2$

In triangle $ABC$, we have $a=BC$, $b=CA$ and $c=AB$ as usual. What is a necessary and sufficient condition for $a^2+c^2=3b^2$ to hold? I created this problem as a generalization of $a^2+c^2=2b^2$ ...
2
votes
1answer
40 views

Construct triangle from three points on base and difference in distances to third vertex

Imagine such a triangle: We know the differences in distances: $\overline{OA} - \overline{BO}$ and $\overline{CO} - \overline{BO}$, as well as the distances between the points on the base: ...
2
votes
1answer
29 views

Similar triangle side lengths given its area and similar triangle side lengths

I've been working through this task in an old textbook and can't figure out where I'm wrong. I suspect my whole approach is wrong. Task says: Given the side lengths of a triangle that are equal to ...
3
votes
1answer
40 views

Triangle geometry: $BC^2+AC^2=n\cdot AB^2$.

I am looking for information regarding which triangles $ABC$ satisfy $BC^2+AC^2=n\cdot AB^2$ for $n=1,2,3,...$. I'm sure that work has already been done in this area since it is a fairly simple ...
0
votes
1answer
62 views

In the figure,What is the ratio of $AE:AD$?

In the figure (not drawn to scale), rectangle $ABCD$ is inscribed in the circle with center at $O$.The length of side $AB$ is greater than side $BC$.The ratio of area of the circle to the rectangle ...
0
votes
1answer
31 views

Simple area and angles of squares and triangels

This is a question APPARANTLY tested on primary 4 and I am in Sec 2,wondering how to do this question....None of my classmates also could finish the question. Question: ABCD and BFGE are squares.AE ...
2
votes
4answers
154 views

Recurrence relation for right-angled triangles stuck-together

Given the image: and that $x_0 = 1, y_0=0$ and $\text{angles} \space θ_i , i = 1, 2, 3, · · ·$ can be arbitrarily picked. How can I derive a recurrence relationship for $x_{n+1}$ and $x_n$? I ...
3
votes
0answers
26 views

Number of triangles created after $n$ folds of a square

My daughter's grade 8 math homework included the following question. We were unable to find an answer, and I think we must have misinterpreted the question, as it seems way too hard. Fold a ...
-3
votes
1answer
25 views

Right triangle - a in terms of c and m [closed]

How can I express a (hypotenuse) in terms of c (one of the sides) and m (the projection of the other side)?
0
votes
1answer
21 views

Angles inequality in acute triangle [duplicate]

Let $\alpha$, $\beta$, $\gamma$ be angles of acute triangle. How to prove that $(\tan(\frac{\alpha}{2}))^2 + (\tan(\frac{\beta}{2}))^2 + (\tan(\frac{\gamma}{2}))^2 \ge 1$? Does left side of equation ...
0
votes
3answers
40 views

Find unknown vertex of triangle given area and other 2 vertices

I need to find the coordinates of the 3rd vertex of a triangle given that I know the other 2 vertices and the area. The triangle is not guaranteed to be of any particular type (right, isosceles, ...
2
votes
2answers
95 views

Geometry: Find angle x in triangle

I have not been able to find a euclidean geometry solution to this, but any other solutions are also appreciated. Let ABC be a triangle with AB=CD and angles as marked in the diagram. Find the ...
0
votes
3answers
52 views

Point inside the area of two overlapped triangles

The question is as simple as that, but I have been trying to figure out an answer (and searching for it) with 0 results. I mean, given two triangles (in 2D) I want to find just a single point which ...
4
votes
7answers
101 views

explaining $|a+b|≤|a|+|b|$ in simple terms

I'm struggling to get to grips with the Triangle Inequalities. The problem is I don't really understand what it means. This is what my lecturer has written in the notes: $$ |a+b|≤|a|+|b|. $$ First of ...
4
votes
2answers
52 views

Bisecting the area and perimeter

In triangle $ABC$, $AB=16$, $AC=15$, and $BC=13$. Point $D$ is on $AB$, and point $E$ is on $AC$ so that $DE$ bisects both the area and perimeter of triangle $ABC$. (In other words, both $DA+AE$ and ...
0
votes
2answers
28 views

Right Triangle Theorem/terminology

Given a line segment $s$, there are exactly two right triangles which have $s$ as a hypotenuse. Is there a name for this theorem? Assuming this line segment lies on a Cartesian plane, how can we ...
0
votes
0answers
15 views

Point distance to verices of triangle with given edges

I would like to find a formulation for the distances between a given point P(x,y), which is inside a general triangle with all edges values provided, to its vertices. Thanks,
1
vote
0answers
28 views

Concave triangle?

I know that in Euclidean and (I think) Spherical geometries don't have concave triangles, but is there any set of axioms that would allow this?
2
votes
4answers
120 views

|a-b|≤|a|+|b| is always true? [duplicate]

I wonder if |a-b|≤|a|+|b| is always true. I think it is true, but I don't see how to prove this mathematically. Thanks.
1
vote
1answer
29 views

Heron's formula to solve a triangle; any faster method?

A triangle has an area of $200cm^{2}$. Two sides of this triangle measure 26 and 40 cm respectively. Find the exact value of the third side. I used Heron's formula to solve this equation, but it ...
0
votes
1answer
21 views

Finding the coordinates of the third point in triangle

How would you find $x$ and $y$ coordinates of the third point in triangle($A$, $B$, $C$), if you know coordinates of $A$ and $B$, and angles at $A$ and $B$?
0
votes
2answers
31 views

Right angled median intersection question

All the information is included in the image. Find the length AB. Only clue I have is that the length CX is 2.5 where X is the perpendicular foot of B which i found out geometrically. However I ...
1
vote
2answers
27 views

Triangle perimeter proof

Given two points, $a, b$ inside a triangle $T$ whose perimeter is L we want to prove that the distance between them: $$|d(a,b)| \leq L. $$ One could easily prove this graphically, ie. the distance ...