For questions about properties and applications of triangles

learn more… | top users | synonyms

2
votes
1answer
28 views

Similar Triangles--Find the measurement of the unknown side [closed]

This is a question I know I got wrong on a final exam in a very easy class for teaching elementary geometry/prep for Praxis II. I actually received a 99% average in the entire course because of the ...
1
vote
1answer
67 views

A geometry problem hinting similarity of triangles .

I recently came across a geometry problem , published in an local magazine(publishing at high school and under graduate level) and was under Difficulty : Hard sub heading. Consider a $\triangle ABC$ ...
0
votes
3answers
36 views

Prove for Pedal & Isosceles triangle.

The tangents at two points $B$ and $C$ on a circle meet at $A$. Let $A_1B_1C_1$ be the pedal triangle of the isosceles triangle $ABC$ for an arbitrary point $P$ on the circle, as shown below. Then ...
2
votes
1answer
119 views

Unique Trianlge Count sequence

Consider a simple graph $G(V,E)$, such that $V = \{1,2,\dots, n\}$. We can define the triangle count of a vertex as follows: $\Delta(v) = $ Number of triangles in the graph such that $v$ is one of ...
0
votes
1answer
28 views

Counting number of points making angle < 90

I have a around 1000 points and 1000 segments in the form of $(x_1, y_1, x_2, y_2)$ meaning the segment starts at coordinate $(x_1, y_1)$ and finishes at $(x_2, y_2)$. For each line i want to know how ...
-3
votes
1answer
43 views

Can you solve this geometric question on triangles? [closed]

In a triangle $ABC$, $D$ is a point on the side $BC$.Given: $AD=10$,$BD=DC=8$ and $BC*AD=6$.What is the length of $BC$? a.$5$ b.$10$ c.$15$ d.$20$ That was asked in a newspaper quiz.
1
vote
2answers
22 views

Get second vertex of isosceles triangle [closed]

Given the equal sides of the triangle and the angle $\theta$ between them as well as the other 2 vertices of the triangle how do I get the second base vertex coordinates. Sorry for my poor drawing. <...
0
votes
4answers
60 views

Triangle - Trapezoid [Geometry]

I'm having trouble with following assignment: "Sides of triangle are $13$, $14$, and $15$. Line parallel to the longest side cuts through the triangle and forms a trapezoid which has perimeter of $39$...
3
votes
3answers
67 views

Prove: In a Triangle, $II_1 = a\cdot \sec \frac{A}{2}$

Prove that $II_1 = a\cdot \sec \dfrac{A}{2}$. $I$ is center of incircle, $I_1$ is center of excircle. What I did is : Drop $ID \perp AB$, & $I_1F \perp AF$ at $F$ So $ID\parallel I_1F$ $\dfrac{...
2
votes
1answer
70 views

Proving Gerretsen's Inequality

Today in class we were shown Gerretsen's inequality: $$16Rr-5r^2\leq s^2 \leq 4R^2+4Rr+3r^2$$ Where $R$, $r$, and $s$ are the circumradius, in radius, and semiperimeter of a triangle. After some ...
0
votes
3answers
57 views

Find third coordinate for a right triangle with 45degree angles

I have a right triangle with two 45degree angles. I know the points for the two coordinates opposite the right angle. I need to calculate the missing point. I have seen similar questions here, but ...
2
votes
2answers
37 views

Prove: $\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$, for circumradius R, inradius $r$, and exradii $r_x$ [closed]

In $\triangle ABC$, prove: $$\frac{r_a}{bc} + \frac{r_b}{ca} + \frac{r_c}{ab} = \frac{1}{r} - \frac{1}{2R}$$ for circumradius $R$, inradius $r$, and exradii $r_a$, $r_b$, $r_c$ in the standard ...
1
vote
1answer
29 views

Geometric proof for properties of Farey sequence

Let $P=(a,c)$ and $P^{'}=(b,d)$ be integral co-ordinates such that $\frac{c}{a}$ and $\frac{d}{b}$ are consecutive terms of Farey sequence. If $O$ is the origin how do I prove no integral co-ordinate ...
0
votes
2answers
54 views

Finding the third side of a triangle given the area

I know the area and the lengths of two sides (a and b) of a non-right triangle. I also know P1 (vertex between a and c) and P2 (vertex between a and b). I already know this much: Perimeter = $ \frac{...
1
vote
1answer
32 views

Area of all triangles involved in a big triangle.

I have a big triangle made up of several small triangle as depicted in picture given below. Suppose, there is one generic triangle of this shape which is formed by joining points arranged in n rows....
-3
votes
0answers
24 views

Sum of Area of Circles. [duplicate]

A circle of radius x cm is inscribed in an equilateral triangle and is tangent at three points. Three smaller circles are inscribed so that they are each tangent to two sides of the triangle and to ...
0
votes
1answer
16 views

Find equal side lengths for isosceles triangle from middle angle and area?

I know that this is a really easy question, but I am looking for the answer to this question: The area of this isosceles triangle is 5cm squared. The angle ABC is 22 degrees. Work out ...
0
votes
1answer
39 views

Problem on circles, tangents and triangles

Let $c_1,c_2,c_3$ be three circles of unit radius touching each other externally. The common tangent to each pair of circles are drawn (and extended so that they intersect) and let the triangle formed ...
0
votes
0answers
34 views

How to solve this $80^\circ$-$80^\circ$-$20^\circ$ triangle ($60^\circ+20^\circ$ and $70^\circ+10^\circ$ variant)? [duplicate]

A friend of mine asked me for help with a math problem and I struggled with this for over an hour. I told him sorry, and I felt bad. It's been bugging me now for hours. I don't even so much care for ...
-1
votes
1answer
18 views

Geometry (ratio of subdivided length in a triangle)

In triangle ABC, label X on AB and Y on AC such that AX : XB = CY : YA = 2 : 1. Extend XY and BC such that they meet at point Z. Find ZB : ZC.
4
votes
1answer
59 views

Find the sides of the triangle.

The triangle with sides $8-15-13$ has a $60^{\circ}$ angle. The triangle with sides $11-35-31$ also has a $60^{\circ}$ angle. Find a triangle $x-y-403$ where $x$ and $y$ are relatively prime positive ...
-4
votes
1answer
32 views

Finding area of triangles [closed]

In a triangle, the average of any two sides is $6 cm$ more than half of the third side , then find the area of the triangle (in$\ cm ^ {2}$)
0
votes
2answers
35 views

Finding Area of the Triangle [closed]

In the figure, the ratio of AD to DC is 3 to 2. If area of $\Delta ABC$ is 40 $cm ^ {2}$ , what is the area of $\Delta BDC $
0
votes
3answers
61 views

If $a^2 + b^2 = c^2$, then $a^3 + b^3 < c^3$, for $a$, $b$, $c$ the sides of a triangle

If $a$, $b$, $c$ are the sides of a triangle where $a^2 + b^2 = c^2$, prove that $a^3 + b^3 < c^3$. I've tried triangle inequality, but I am stuck.
0
votes
2answers
58 views

How to find the tangency condition for this circle geometry problem?

Suppose I have a circle $C$ of radius $1$, and I have a chord of this circle, of given length $l$. The chord makes a known angle $\theta$ with the tangent to the circle. I position a smaller circle $...
3
votes
2answers
137 views

If the sides of a triangle satisfy $(a-c)(a+c)^2+bc(a+c)=ab^2$, and if one angle is $48^\circ$, then find the other angles.

In triangle $ABC$ one angle of which is $48^{\circ}$, length of the sides satisfy the equality: $$(a-c)(a+c)^2+bc(a+c)=ab^2$$ Find the value in degrees the other two angles of the triangle. I ...
1
vote
2answers
57 views

Construct the triangle with given points and lines

On the following picture you see the excersice handed to us. Construct triangle ABC when you know that x is the line that contains points B and C, line z is the median that goes trough point A and ...
1
vote
1answer
26 views

3D Geometry concurrency problem

$ABCD$ is a tetrahedron. Let $K$ be the center of the incircle of $CBD$. Let $M$ be the center of the incircle of $ABD$. Let $L$ be the centroid of $DAC$. Let $N$ be the centroid of $BAC$. Suppose $...
1
vote
3answers
64 views

For $\triangle ABC$, prove $( \sin A + \sin B )( \sin B +\sin C )( \sin C + \sin A) > \sin A \sin B \sin C$

In $\triangle$ ABC, prove that $$( \sin A + \sin B )( \sin B + \sin C )( \sin C + \sin A) > \sin A \sin B \sin C$$ I have tried the formula A.M.- G.M. relation with $\sin A$, $\sin B$, and $\...
0
votes
1answer
29 views

Triangle wave equation

I have a triangle wave equation represented as $$ y = \dfrac{A \cdot \left(P - \lvert\;\left(x \mod (2 \cdot P) \right) - P \;\rvert\right)}{P} $$ where $A$ is the amplitude and $P$ is half of the ...
1
vote
3answers
108 views

Minimizing $\cot^2 A +\cot^2 B + \cot^2 C$ for $A+B+C=\pi$

If $A + B + C = \pi$, then find the minimum value of $\cot^2 A +\cot^2 B + \cot^2 C$. I don't know how to solve it. And can you please mention the used formulas first. What I can see is that if one ...
1
vote
0answers
38 views

split a rectangle with triangles into polygons as uniformly as possible

Given a rectangle $A$ and $n$ triangles $\{B_1,B_2,...,B_n\}$, I put the triangles inside $A$, at least one vertex of each triangle is not outside $A$ (inside $A$ or on the edge of $A$). So that A is ...
0
votes
0answers
35 views

Discovering length of line

I'm attempting to work out length of BD from below diagram : The length of BD is -2 +- some value. But since I do not know the y co-ordinate of B can the length of BD be determined from ...
0
votes
1answer
40 views

Prove concurrency in a triangle

If a circumference cuts a triangle $ABC$ at its sides $BC$, $CA$ and $AB$ at points $P, P'; Q, Q'; R, R'$; respectively (so twice on each side, and if $AP, BQ$ and $CR$ are concurrent (intersect at a ...
2
votes
3answers
172 views

Prove triangle similiarity by given expression

I am working on the following problem, but I can't seem to figure it out. The length of the sides in the triangle $T_1$ are $a_1$, $b_1$ and $c_1$. The length of the sides in the triangle $T_2$ ...
0
votes
2answers
53 views

Angle Between Two Tangents

In the picture below, the angle $AOB$ is $\delta \theta$, and then it is deduced that the angle between the two tangents is the same from the fact that the angles in a quadrilateral add up to $2 \pi$. ...
-1
votes
1answer
35 views

Question from triangles [closed]

in 🔺ABC, P and Q are points on sides AB and AC respectively, such that PQ||BC . If AP=2.4 cm, AQ=2cm , QC=3 cm and BC=6cm , find AB and PQ?
0
votes
1answer
38 views

Working out length of side of triangle?

I'm taking mooculus course from https://mooculus.osu.edu/exercises/linearTriangles1 and am given following problem : What is the intuition of the hint : 'length of DA = abscissa of D minus abscissa ...
1
vote
1answer
39 views

Derive a relation between angles A,B and C

Derive a relation between angles A,B and C (do not use other angles in the final relation): I have tried to use two theorems in triangles(external angle and complement angles),but no success! It ...
0
votes
2answers
113 views

What is the size of the angle $\angle AMC$? [duplicate]

Suppose we have a triangle $\triangle ABC$ where the size of two angles are given: $\angle B=15^\circ$ and $\angle C=30^\circ$. We draw the median $AM$, so now what is the size of angle $\angle AMC$? ...
0
votes
4answers
51 views

How do I find the height of a triangle when it is tilted downwards at one end?

In the first pic, it is shown that the height of the triangle is $1.5$ m. In the second pic, the point $C$ is moved to point B. How do I find height $h$ so that the perpendicular height of the ...
1
vote
2answers
46 views

What should I do further?

I came across simple question, The length of all sides of a $\triangle{ABC}$ are in integral units. If length of $AB=10$ and $AC= 15$ then the number of distinct possible values of $BC$ is finite. We ...
10
votes
1answer
2k views

Triangle dissection, no shared edges

It's possible to divide a triangle into smaller triangles such that no edge lengths are shared. Alternately, no two internal triangles share two vertices. The top three are the known simplest ...
0
votes
1answer
31 views

Trigonometric problem regarding a tower

The angle of elevation of a tower, $CD$, from a point $A$ due East of the tower is 45°. From a point $B$ due south of $A$, the angle of elevation is 30°. The distance from $A$ and $B$ is 400 metres. ...
2
votes
4answers
95 views

find the measure of $AMC$

if $M$ is the midpoint of $BC$ then find the measure of $AMC$. I tried to use the angles to find $AMC$ but I don't know how to use that $M$ is the midpoint of $BC$.
1
vote
1answer
56 views

Triangle Inequality for $\|x\|_{\infty}$

I have to show the triangle inequality for $\|x\|_{\infty}$. I'm not sure, if estimate is correct. To show: $\|x+y\|_{\infty} \le \|x\|_{\infty}+\|y\|_{\infty}$ Let $x \in \mathbb{R}^n$ and $\|x\|_{\...
1
vote
3answers
42 views

Drawing triangle

follow the picture: $m$,$p$ and $Q$ are midpoints of segments we want to draw the triangle and we only have the lengh of $AM$,$BQ$ and $CP$ How to draw the triangle?
0
votes
1answer
84 views

A triangle in a square

The following quadrilateral is a square also there are some known angles.prove that The segments of the inner triangle are equal. My Attempt:If we name the inner point $O$ then two triangles $AOD$ ...
0
votes
1answer
10 views

Finding the circumcentre

Suppose angle ACB is 90°, why is p the circumcentre of triangle ACB? I can only proove that RP is the perpendicular bisector but what is it to do with angle ACB?
0
votes
0answers
30 views

An geometry problem. proved that the circles inscribed in triangle ABD&CAD are tough each other.

The inner circle of triangle $ABC$ touches $BC$ at $D$ . Show that the circles inscribed in triangles $ABD$ and $CAD$ touch each other.