For questions about properties and applications of triangles

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3answers
34 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
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2answers
31 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 $r_x$

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 ...
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1answer
22 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 ...
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2answers
49 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
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1answer
23 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....
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0answers
23 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
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1answer
15 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 ...
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1answer
36 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 ...
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0answers
33 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 ...
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1answer
15 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.
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1answer
56 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 ...
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1answer
31 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$\ cm ^ {2}$)
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2answers
33 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 $
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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.
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2answers
52 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
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2answers
127 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 ...
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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 ...
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1answer
23 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 $...
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0answers
36 views

How do I find the distance from the vertex to some point inside an isosceles triangle? [closed]

The figure I am working with is part of a circle. I was given the radius which is 45.4 as well as the angle which is 0.1 radians or 5.7 degrees. I also know that the correct answer is 32.049.... I ...
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3answers
58 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 $\...
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1answer
27 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 ...
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3answers
102 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 ...
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0answers
36 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 ...
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0answers
34 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
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1answer
39 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 ...
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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 $T_2$ ...
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2answers
51 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 $2 \pi$. ...
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1answer
34 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?
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1answer
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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 ...
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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
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2answers
107 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 $\angle AMC$? ...
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4answers
50 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 ...
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2answers
43 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 ...
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0answers
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 ...
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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. ...
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4answers
89 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 $BC$.
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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\|_{\...
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3answers
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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?
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1answer
81 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
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1answer
9 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?
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0answers
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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.
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1answer
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equilateral triangle and inscribed circle

Let ABC be an equilateral triangle. let D be a random point on BC. Let I_1 and I_2 be the incenters in ABD and ADC. Let O be the circumcentre of AI_1I_2. Prove that OD is perpendicular to BC.
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1answer
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In the triangle $ABC$, $AB>AC$. Show that $BE>CF$ where $E$ and $F$ are the midpoints of $AC$ and $AB$

In the triangle $ABC$, $AB>AC$ and $E,F$ are the midpoints of $AC$ and $AB$. Show that $BE>CF$. I going this problem in the excursion of mathematics book. I try it very much but can't able to ...
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3answers
29 views

Prove that $AH^2+BC^2=4AO^2$

Prove that $AH^2+BC^2=4AO^2$, where $O$ is the circumcentre and $H$ is the orthocentre of the triangle $ABC$.
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3answers
42 views

Area of a triangle whose vertices lie on a parallelogram

In the parallelogram $ABCD$, $X$ and $Y$ are the midpoints of $BC$ and $CD$. Then prove that $$Ar(\triangle AXY) = \frac {3}{8} Ar(ABCD)$$ My Attempt : Construction; Joining $BY$ and $AC$, I got ...
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1answer
33 views

If AO=AH. prove that angle A=60 degree.

$ABC$ is a triangle with the circumcenter $O$ and orthocenter $H$. If $AO=AH$, prove that $m(\hat A)= 60^\circ$. Also, if $H$ lies on the circumcircle of $BOC$, prove that $m(\hat A)= 60^\circ$.
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1answer
33 views

Finding points on a right triangle

I have the points A, B and C. I also have the angle alpha between AB and BD or BE, and I know l = |BD| or |BE|. But how can I find D or E?
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2answers
165 views

Triangle in perspective to a given triangle but similar to another

Is it always possible to construct a triangle that is in perspective to a given triangle and have it also be similar to a different given triangle? If you create a triangle in perspective to another, ...
0
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0answers
20 views

P is any point inside a triangle . prove that s<AP+BP+CP<2s [duplicate]

P is any point inside a triangle ABC. The perimeter of the triangle AB+BC+CA=2s. Prove that s<AP+BP+CP<2s.
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0answers
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When does there exist a point with a given ratio of distances to the vertices of a triangle?

I have the triangle ABC and an unknown point P not necessarily inside the triangle. Also, I have three lengths (...