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Questions tagged [triangles]

For questions about properties and applications of triangles.

2
votes
1answer
34 views

Clarification in proof of perpendicular bisectors meeting at a point

Context: My crude drawing from Paint to illustrate triangle OAB: My working: $\begin{align*} (z-\frac{1}{2}(x+y)) \cdot (y-x) &= z\cdot(y-x) + \frac{1}{2}(-x-y) \cdot (y-x)\\ &= z\cdot y -z\...
0
votes
1answer
15 views

Given a set of points, find maximum area of triangle

Given a finite set of 2-d points, I need to find the maximum area of triangle formed. My solution steps : Take mean of points , lets call it (x_m, y_m). Take 3 most distant points from (x_m, y_m). ...
0
votes
1answer
41 views

Geometry triangle inside circle [on hold]

Triangle $\triangle ABC$ is inscribed in a circle. $D$ is a point on $AC$. $BD$ is angle bisector of $\angle B$. $O$ is the center of the circle then find $\angle ADO$ if $\angle A=20°$ and $AB=AC.$
2
votes
3answers
47 views

Two circles inscribed in isosceles triangle

$$\triangle ABC :AC =BC$$ $$P \in AB$$ $k(O;r)$ is inscribed in $\triangle APC$; $k_2(O_2; r_2)$ is inscribed in $\triangle BPC$; $D, G$ are points of contact of the circles with $CP$. Show ...
5
votes
2answers
159 views

Inscribed circle in right-angled triangle

In right-angled $\triangle ABC$ with catheti $a = 11\,\text{cm}, b=7\,\text{cm}$ a circle has been inscribed. Find the radius and altitude from $C$ to the hypotenuse. I found that the hypotenuse is $...
0
votes
1answer
22 views

Find 1 point from 3 Angles, 2 Points and 1 unit vector

i have the following problem: Given are points P1 and P2. I also have the direction, in the image noted as D, given as a unit vector. Additionally, I have the angles alpha and gamma and I know that ...
1
vote
4answers
56 views

About two inscribed circles in a right triangle

In $\triangle ABC$, $H$ is the foot of prependicular from $A$ to $ {BC}$. $\angle A = \frac{\pi}{2}$. $O_1, O_2$ are the centers of inscribed circles of $\triangle AHB$, $\triangle CHA $ respectively. ...
3
votes
4answers
76 views

Show that a triangle is right angled

In $\triangle ABC$ (not isosceles) $CH$, $CL$ and $CM$ are respectively height, bisector and median. Show that $\angle ACB = 90^\circ$ if and only if $\angle HCL = \angle MCL$. I think that I have to ...
6
votes
1answer
41 views

Proof that line $HG$ bisects the perimeter of $ABC$

Question. From this picture, $D$ and $E$ are excenters of $ABC$, and $G$ and $H$ are midpoints of $AB$ and $KL$. Prove that $HG$ bisects the perimeter of $ABC$. In other words, prove that $AX=CX+CB$....
7
votes
3answers
187 views

Prove that $BD$ bisects $\angle ABC$

Given that $\triangle ABC$ is an isosceles right triangle with $AC=BC$ and angle $ACB=90°$. $D$ is a point on $AC$ and $E$ is on the extension of $BD$ such that $AE$ is perpendicular to $BE$. If $AE=\...
-1
votes
0answers
14 views

Triangles -geometry

ABC is a triangle right angled at A ,AP and AQ meet BC or BC produced in P and Q and are equally inclined to AB .Show that BP:BQ=PC:CQ I think we have to use the property of angle bisectors ,that is ...
0
votes
1answer
17 views

I want to understand the triangle inequality about contraction operator

I read the Chow's paper, Multigrid algorithms and complexity results, https://dspace.mit.edu/handle/1721.1/14254 I have a question on page 42. Let me write the Lemma 2.4.3 on the page. Lemma 2.4.3 ...
0
votes
0answers
6 views

Incenter divides each bisector at the same ratio

Prove that if the incenter divides each of the bisectors at the same ratio then the triangle is equilateral I've tried using the bisectors theorem but apparently it doesn't apply here
0
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0answers
18 views

Geometry - Straight lines and triangles

Let ABCD be a quadrilateral and X be the midpoint of BD .Prove that the area of AXCB is one half area of ABCD. The problem seems really easy ,however i am not able to prove it for any quadrilateral ,...
-2
votes
0answers
19 views

1. Straight lines and triangles

If S is the circumcentre of a triangle ABC and D,E,F are the feet of the altitudes of triangle ABC then prove SB is perpendicular to BF. I have seen solutions to this problem using cyclic ...
0
votes
1answer
27 views

Straight lines and triangles

AB and CD are two fixed straight lines and a variable straight line cuts them at X and Y respectively .The angular bisectors of angle AXY and angle CXY meet at P.Find the locus of P. My answer is P ...
0
votes
1answer
16 views

Corresponding angles of $\square ABCD$ and $\square PQRS$ are equal, $AB=PQ$, $CD=SR$, $AD\not\parallel BC$; prove the quadrilaterals congruent

If two quadrilaterals $ABCD$ and $PQRS$ have angles $A$, $B$, $C$, $D$ equal to angles $P$, $Q$, $R$, $S$ respectively and $AB=PQ$, $DC=SR$, and if $AD$ is not parallel to $BC$, prove that the ...
0
votes
0answers
19 views

Find a point on a circle which contains a rectangle using another point, the angle between the two and the rectangle's dimensions

I'm trying to construct a mathematical formula that will calculate a point (x,y) on a circle which contains a rectangle with a width ...
1
vote
0answers
32 views

Probability of obtuse triangles formed on a circle

There are 16 equally spaced points on the circumference of a circle. If 3 points out of these 16 points are selected randomly, What is the probability that they will form an obtuse angled triangle?
2
votes
4answers
169 views

Find the length x such that the two distances in the triangle are the same

I have been working on the following problem Statement Assume you have a right angle triangle $\Delta ABC$ with cateti $a$, $b$ and hypotenuse $c = \sqrt{a^2 + b^2}$. Find or construct a point $D$ ...
0
votes
1answer
88 views

Find $P$ in the plane of $\triangle ABC$ that minimizes $a\,AP^2+b\,BP^2+c\,CP^2$

From KMO: Let $\triangle ABC$ be a triangle. Find a point $P$ in the plane of $\triangle ABC$ such that $a\,AP^2 + b\,BP^2 + c\,CP^2$ is minimum. How can I solve this problem by using the basic ...
8
votes
1answer
786 views

Understanding Ceva's Theorem

In Ceva's Theorem, I understand that $\frac{A_{\triangle PXB}}{A_{\triangle PXC}}=\frac{BX}{CX}=\frac{A_{\triangle BXA}}{A_{\triangle CXA}}$. I would like clarification in understanding the following ...
0
votes
0answers
13 views

Producing orientation (yaw)

I am working with a car and trailer model which has a GPS sensor attached and I am looking for ways to calculate it's yaw (orientation). One way I am working on is two resolve both the car and ...
1
vote
1answer
23 views

$n$ incongruent triangles with integer sides and perimeter $n$

What is the only positive integer $n$ such that there are exactly $n$ incongruent triangles with integer sides and perimeter $n$? I have found that the answer to the above problem is $n = 48$. I know ...
-1
votes
2answers
52 views

How can I find the sum of the angle $AMB$, angle $ANB$ and the angle $ACB$? [closed]

How can I find the sum of the $\angle AMB, \angle ANB$ and the $\angle ACB$? In triangle $ABC$, $\angle ABC =90^\circ$. $BC$ is divided in $3$ parts such that $BM=BN=NC$. And also $AB=BM$. Here are 2 ...
0
votes
2answers
57 views

One angle in a triangle is twice the other, find the relationship among the sides

$\angle A$ is twice as $\angle B$. I need to find formula that describes relationship among $a,b,c$. It should be a function of only $a,b,c$. I tried using cosine law and sine law but no result.
2
votes
2answers
69 views

$ABC$ is an arbitrary triangle. Why are three points $X, Y, Z$ located on a line?

In this picture, $ABC$ is an arbitrary triangle and $O$ is the center of the Inclining circle of the triangle $ABC$. We extend the sides of $AB$ and we draw an arbitrary line from the center of the ...
1
vote
2answers
41 views

An angle bisector of length $4$ units, creates a $6$ unit line segment on the base of the triangle. What are possible integer values of its side?

What is the sum of the possible integer values of $x$ My Solution I know that $\angle DAC \lt 90^o$ because it is an angle bisector, therefore: $$4^2+x^2 \lt 6^2 \quad \quad 6-4 \lt x \lt 6+4$$ ...
1
vote
1answer
53 views

Is there a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle?

Can anyone provide a geometric proof of $\frac1r = \frac{1}{h_a} + \frac{1}{h_b} + \frac{1}{h_c}$ in a given triangle, if there is one? Here, $h_i$ refers to the distance from the side $i$ to the ...
4
votes
1answer
48 views

Geometric problem about ratios of line segments: How to transform the limiting case method to a rigorous answer?

In $\triangle ABC$, $D, E, F, G$ are points on the sides of the triangle such that $BD:DE:EC=1:2:3$, $AF:CF=1:1$, and $AG:BG=2:3$. Find the ratio $FH:DH$. My classmate has come up with a brilliant ...
1
vote
1answer
64 views

Problem about incentre of a triangle.

Through the incentre $I$ of triangle $ABC$ a straight line is drawn intersecting $AB$ and $BC$ at points $M$ and $N$, respectively, in such a way that the triangle $BMN$ is acute- angled. On the ...
4
votes
5answers
141 views

Finding Angle using Geometry

In an equilateral triangle $ ABC $ the point $ D $ and $ E $ are on sides $ AC $ and $AB$ respectively, such that $ BD $ and $ CE $ intersect at $P$ , and the area of the quadrilateral $ ADPE $ is ...
1
vote
4answers
50 views

Are angles of a triangle always positive?

If a question says "$A,B,$ and $C$ are the angles of a triangle". Can I assume $A,B,$ and $C$ to be positive and its properties such as arithmetic mean greater equal to geometric mean? Also angle ...
1
vote
1answer
33 views

Geometric inequality

Given Triangle ABC, let D be mid point of BC and let E and F be two point on sides AB and AC respectively such that angle EDF is right angle (90°). Prove that BE + CF > EF. I was able to do it by ...
1
vote
2answers
39 views

Problem-solving with ratios and similar triangles.

The following question is designed to test problem-solving and reasoning skills with ratios and similar triangles. Without using Pythagoras Theorem or Trigonometry this question is supposed to be ...
0
votes
1answer
23 views

Simple formula for radius of circumcircle from coordinates of 3 points

Given $A=(a_1,a_2), B=(b_1,b_2), C=(c_1,c_2)$, is there a simple formula to express the radius of the circumcircle of $ABC$? Note that you could compute the radius from the sidelengths as $\frac{abc}{...
-1
votes
1answer
18 views

Proof for similarities between two triangles.

We know that if the angles of two triangles are similar, then their sides are proportional. I get the idea. Now, can it be proven rigorously?
-2
votes
1answer
32 views

How to find the angle? [closed]

The altitudes of a triangle ABC meet at point H. You know, AB=CH then determine the value of angle BCA . Couldn't get the angle at all... Please, help.
1
vote
2answers
35 views

Find a point on segment of triangle that cuts 90 degrees from other triangle point

For any triangle, how can I find the point D, knowing A,B and C and that the segment CD must be perpendicular to AB ? Example :
2
votes
1answer
90 views

Problem involving triangle. Find $x$ in the figure

I need to find $x$ in the triangle above. I tried to do basic things, like sum of a triangle's internal angles $= 180^\circ$ but I only found $2$ equations for $3$ variables Any help is appreciated
1
vote
1answer
65 views

$h^2 = x^2 + (x+1)^4$

This is a question I had of finding an exact value of $h$ as I was interested in the process and techniques used. The question was one to do with Pythagoras and the math shown is the question in $a^2 +...
-3
votes
1answer
57 views

find a certain ratio of areas in a triangle where other areas are given [closed]

Given : [BDE]=8, [BDC]=12, [CDF]=9 where [.] is the area of the respective triangles. Find ratio [AFD]/[AED] from the figure. NOTE:-Question may have errors please just edit that or comment the ...
3
votes
2answers
72 views

Successive prime numbers and triangles

Except the triplet $(2,3,5)$, I am wondering if there exists another triplet of successive prime numbers such that they do not form a triangle (each prime number corresponds to a side of the triangle)....
1
vote
2answers
54 views

Finite element heat equation on a single simplex?

I am currently trying to learn the finite element method. Ultimately, I want to solve the heat equation in arbitrary dimensions. For the purpose of this question, however, assume that I am interested ...
-2
votes
1answer
43 views

Right Triangle Angle bisectors and Proofs [closed]

I’ve been looking at this question for hours. Anyone know how to solve it? Here is the question diagram to help. Any help would be appreciated! https://i.stack.imgur.com/Z313k.jpg I’ve done this so ...
0
votes
6answers
69 views

What's the length of the segment? [closed]

The triangle is an Equilateral. If $\angle BDP = 90^\circ, \angle PEB = 90^\circ, \angle DBP = 22.5^\circ$ and $\angle PBE = 37.5^\circ, BP = 4\sqrt 3$, and each side of the triangle is $8$, then what'...
3
votes
2answers
1k views

What is the value of $\alpha$ and $\beta$ in a triangle?

On triangle $ABC$, with angles $\alpha$ over $A$, $\beta$ over $B$, and $\gamma$ over $C$. Where $\gamma$ is $140^\circ$. On $\overline{AB}$ lies point $D$ (different from $A$ and $B$). On $\...
1
vote
1answer
34 views

If ratio of sides of two triangles is constant then the triangles have the same angles

If $\triangle ABC$ and $\triangle A'B'C'$ are a pair of triangles such that $$\dfrac{|AB|}{|A'B'|}=\dfrac{|BC|}{|B'C'|}=\dfrac{|AC|}{|A'C'|}$$ then $$\triangle ABC \sim \triangle A'B'C'$$ I have ...
0
votes
1answer
51 views

Can't understand this sentence , please draw this triangle [closed]

Each side of an arbitrary triangle is divided into three equal parts, a small triangle is cut off at each vertex of the triangle along the line which connects the two adjacent division points on the ...
0
votes
2answers
29 views

Let $AD$ be the angle bisector of angle $A$ in $\Delta ABC$. Prove that $BD = BC$ . $\frac{AB}{AB + AC}$

Let $AD$ be the angle bisector of angle $A$ in $\Delta ABC$. Prove that $$BD = BC \cdot \frac{AB}{AB + AC}$$ Hello, I was doing some geometry and got stuck in this question. I tried using the angle ...