For questions about properties and applications of triangles

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3
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4answers
123 views

How to prove that triangle inscribed in another triangle (were both have one shared side) have lower perimeter?

This question looks really simple, but to my (and my co-workers) frustration we were not able to prove this in any way. I tried all triangle formulas known to me but I feel I'm missing the point, and ...
3
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1answer
2k views

calculating the Fermat point of a triangle

Is there any algorithm by which one can calculate the fermat's point for a set of 3 points in a triangle? a fermat's point is such a point that the sum of distances of the vertices of the triangle to ...
2
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1answer
65 views

Two inequalities in a triangle

I'm trying to prove that in a triangle with side lengths $a,b,c$, median lengths $m_a, m_b, m_c$ and circumdiameter $D$ the following inequality holds: $$ ...
2
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3answers
2k views

How many triangles can be created from a grid of certain dimensions?

How would you determine how many non-degenerate triangles can be drawn by connecting points in a $5 \times 5$ grid?
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1answer
3k views

Related rates: Find dA/dt of triangle, given d(theta)/dt — Can't come to textbook answer

The question: ABC is a triangle in which the lines $\overline {AB} = 20cm$, $\overline {AC} = 32cm$ and $\angle BAC = \theta$. If $\theta$ is increasing at the rate of 2° per minute, determine the ...
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vote
1answer
222 views

Ravi substitution in inequalities

There is a well-known substitution for proving geometric inequalities: If $a,b,c$ are the side lengths of a triangle, then in an inequality involving $a,b,c$ it is possible to replace $a,b,c$ by ...
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vote
5answers
266 views

Find the length of a leg of a right triangle, given the area and the length of the other leg

The length of one leg of a right triangle is $(x - 6)$ centimeters, and the area is $(\frac12 x^2 - 7x + 24)$ square centimeters. What is the length of the other leg? I think the equation that I need ...
0
votes
1answer
86 views

Area of convex n-gon using triangles

Suppose we have a convex $n$-gon and a point inside the $n$-gon or on the sides of the $n$-gon, and suppose one extended lines from all the vertices of the $n$-gon to make $n$ triangles with two of ...
0
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2answers
56 views

Linear algebra - find all possible positions of the third corner?

An equilateral triangle lies in the plane $x + y - z = 1$ and corners in points $(1, 1, 1)$ and $(2, 1, 2)$. Determine all possible positions of the third corner?
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5answers
311 views

Prove that $\angle FGH = \angle GDJ$

Let $FGH$ be a triangle with circumcircle $A$ and incircle $B$, the latter with touchpoint $J$ in side $GH$. Let $C$ be a circle tangent to sides $FG$ and $FH$ and to $A$, and let $D$ be the point ...
25
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1answer
598 views

Can Three Equilateral Triangles with Sidelength $s$ Cover A Unit Square?

A previous question on the site asked for a short proof of the fact that three equilateral triangles with unit side length cannot be arranged to cover a square with unit side lengths. Given the truth ...
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votes
4answers
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Proving Stewart's theorem without trig

Stewart's theorem states that in the triangle shown below, $$ b^2 m + c^2 n = a (d^2 + mn). $$ Is there any good way to prove this without using any trigonometry? Every proof I can find uses the ...
18
votes
4answers
251 views

A triangle determinant that is always zero

How do we prove, without actually expanding, that $$\begin{vmatrix} \sin {2A}& \sin {C}& \sin {B}\\ \sin{C}& \sin{2B}& \sin {A}\\ \sin{B}& \sin{A}& \sin{2C} \end{vmatrix}=0$$ ...
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6answers
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How many triangles are there?

The question is how many triangles are there in the following picture? I have thought to solve it by creating a formula based on the angles of the lines starting from the bottom of each side. I ...
18
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2answers
685 views

Is ABC an equilateral triangle

In the figure, AD=BE=CF. Moreover, DEF is an equilateral triangle. Must ABC be equilateral?
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2answers
502 views

Do there exist an infinite number of 'rational' points in the equilateral triangle $ABC$?

Let's call a point $P$ which satisfies the following condition 'a rational point'. Condition: Each distance $PA, PB, PC$ from a point $P$ to three vertices $A, B, C$ of an equilateral triangle $ABC$ ...
14
votes
2answers
114 views

The inequality $\frac{MA}{BC}+\frac{MB}{CA}+\frac{MC}{AB}\geq \sqrt{3}$

Given a triangle $ABC$, and $M$ is an interior point. Prove that: $\dfrac{MA}{BC}+\dfrac{MB}{CA}+\dfrac{MC}{AB}\geq \sqrt{3}$. When does equality hold?
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2answers
28k views

How to find surface normal of a triangle

If I have a triangle with $3$ points $P_1, P_2,$ and $P_3$, each with $x, y,$ and $z$ coordinates, how do I find the surface normal $N$ in $x, y,$ and $z$ such that $$N_x+N_y+N_z = 1$$ I'm looking ...
11
votes
2answers
5k views

Maximum area of a square in a triangle

I want to calculate the area of the largest square which can be inscribed in a triangle of sides a, b, c. The "square" which I will refer to, from now on, has all its four vertices on the sides of the ...
10
votes
2answers
288 views

A geometric inequality, proving $8r+2R\le AM_1+BM_2+CM_3\le 6R$

Here, $AM_1$ is the angle bisector of $\angle A$ extended to the circumcircle and so on. $R$ is the circumradius and $r$ is the inradius, respectively. I have to prove that: $$8r+2R\le ...
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votes
4answers
2k views

Find an angle in a given triangle

$\triangle ABC$ has sides $AC = BC$ and $\angle ACB = 96^\circ$. $D$ is a point in $\triangle ABC$ such that $\angle DAB = 18^\circ$ and $\angle DBA = 30^\circ$. What is the measure (in degrees) of ...
16
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1answer
274 views

Under what conditions will the rectangle of the Japanese theorem be a square?

In geometry, the Japanese theorem for cyclic quadrilaterals states that the centers of the incircles of certain triangles inside a cyclic quadrilateral are vertices of a rectangle. Question. Under ...
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6answers
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how to calculate area of 3D triangle?

I have coordinates of 3d triangle and I need to calculate its area. I know how to do it in 2D, but don't know how to calculate area in 3d. I have developed data as follows. ...
11
votes
1answer
119 views

Is there a name for the recursive incenter of the contact triangle?

Recently, I became aware that there are many more triangle centers than the four I learned about in school. This reminded me of a thought I had when I first learned about the incenter: what point ...
5
votes
3answers
9k views

Triangle inequality for subtraction?

Is the following inequality(that looks like the triangle inequality) valid: $|a - b| \leq |a| - |b|$ Why?
18
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4answers
2k views

probablity of random pick up three points inside a regular triangle which form a triangle and contain the center

what is the probablity of random pick up three points inside a regular triangle which form a triangle and contain the center of the regualr triangle the three points are randomly picked within the ...
11
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2answers
4k views

Equilateral triangle inscribed in a triangle

Consider triangle ABC. Point D is on AC, E is on BC and F is on AB. Given that triangle DEF is equilateral and that segments CD, BE and AF are equal in length, prove that triangle ABC must also be ...
9
votes
1answer
323 views

The case of Captain America's shield: a variation of Alhazen's Billard problem

I'm sure a lot of you are acquainted with Alhazen's Billiard problem, which involves finding the point on the edge of a circular billiard table at which a cue ball at a given point must be aimed in ...
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2answers
377 views

How to prove that $\frac{r}{R}+1=\cos A+\cos B+\cos C$?

How do we prove that for any triangle this holds: $$\frac{r}{R}+1=\cos A+\cos B+\cos C$$ I can use this beautiful identity to prove several geometric inequalities, but I have no idea how to prove the ...
4
votes
3answers
141 views

$PA^2\sin A+PB^2\sin B+PC^2\sin C$ is minimum if P is the incenter.

Let $ABC$ be a triangle and $P$ is a point in the plane of the triangle $ABC$.If $a,b,c$ are the lengths of sides $BC,CA,AB$ opposite to angles $A,B,C$ respectively then prove that $PA^2\sin ...
4
votes
2answers
778 views

Prove that the centre of the nine-point circle lies on the midpoint of the Euler line

In $\Delta ABC$, $AD, BE, CF$ are the altitudes and $\Delta A'B'C'$ is the medial triangle. $K, L, M$ are the midpoints of $AH, CH, BH$. Consider the nine-point circle with centre $G$ (not to be ...
3
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1answer
146 views

I need help with this geometry question.

Let $ABC$ be a triangle with $AB=AC$. If $D$ is the midpoint of $BC$, $E$ is the foot of the perpendicular drawn from $D$ to $AC$ and $F$ the mid-point of $DE$, prove that $AF$ is perpendicular to ...
2
votes
2answers
126 views

$\frac{AB}{A'B'}+\frac{BC}{B'C'}+\frac{CA}{C'A'} \geq 4 \left(\sin{\frac{A}{2}}+\sin{\frac{B}{2}}+\sin{\frac{C}{2}}\right). $

Let be a circle inscribed in the triangle $\triangle ABC$ wiht the center $I$. The intersection of the circle with $AI$ is $A'$, with $BI$ is $B'$ and with $CI$ is $C'$. Prove that: ...
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4answers
1k views

Does a triangle always have a point where each side subtends equal 120° angles?

Is there a point $O$ inside a triangle $\triangle ABC$ (any triangle) such that the angle $\angle{AOB} = \angle{BOC} = \angle{AOC}$? What do we call this point?
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6answers
2k views

Finding the largest triangle inscribed in the unit circle

Among all triangles inscribed in the unit circle, how can the one with the largest area be found?
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2answers
1k views

The incenter and Euler line.

It seems well known that the incenter of a triangle lies on the the Euler line if and only if the triangle is isosceles (or equilateral, but that is trivial). Searching the internet, I could not find ...
5
votes
1answer
6k views

How many triangles with integral side lengths are possible, provided their perimeter is $36$ units?

How many triangles with integral side lengths are possible, provided their perimeter is $36$ units? My approach: Let the side lengths be $a, b, c$; now, $$a + b + c = 36$$ Now, $1 \leq a, b, c ...
3
votes
1answer
116 views

Proving a tough geometrical inequality, with equality in equilateral triangles.

For any triangle with sides $a ,b, c$ prove or disprove (1) and (2) : $$\sum_\mathrm{cyc} \frac{1}{\frac{(a+b)^2-c^2}{a^2}+1}\ge \frac34$$ Equality in (1) holds if and only if the triangle is ...
10
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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 ...
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3answers
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What is the flaw in this proof that all triangles are isosceles?

What is the flaw in this "proof" that all triangles are isosceles? From the linked page: One well-known illustration of the logical fallacies to which Euclid's methods are vulnerable (or at least ...
5
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3answers
939 views

Largest Triangle with Vertices in the Unit Cube

How would one find a triangle, with vertices in or on the unit cube, such that the length of the smallest side is maximized? And what is that length? A lower bound for the length is $\sqrt{2}$, by ...
5
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1answer
3k views

Find the point in a triangle minimizing the sum of distances to the vertices

Given a triangle in a plane with vertices A, B, C, find the point T that minimizes the sum of distances between A-T, B-T, and C-T. I can experimentally determine this point by sampling the space and ...
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7answers
216 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 ...
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2answers
1k views

how many rectangles in this shape

I've learned in my high school the solution to such riddle: How many rectangles are there in this shape: the solution is through combinations: in this shape is a $5\times 6$ grid so the number of ...
3
votes
3answers
172 views

Find out the angle of <ABC

Help me to solve it please.how could it be done.I tried but nothing comes out.Help me please
2
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1answer
379 views

Existence of Gergonne point, without Ceva theorem

The intersection at one point (called Gergonne point) of the lines from vertices of a triangle to contact points of the inscribed circle can be proved immediately using Ceva's theorem. Is there a ...
2
votes
3answers
9k views

Calculate coordinates of 3rd point (vertex) of a scalene triangle if angles and sides are known.

I am writing a program and I need to calculate the 3rd point of a triangle if the other two points, all sides and angles are known. ...
0
votes
4answers
71 views

How to prove using Plane Geometry that Centroid divides in ratio $2$:$1$ [closed]

In $\Delta ABC$ Can any one give me a hint to Prove that the centroid $G$ divides $A$ and Mid point of $BC$ in the ratio $2$:$1$ Using only Plane Geometry.
0
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3answers
149 views

A triangle with vertices on the sides of a square, with one at a midpoint, cannot be equilateral

Let A, B, C, D, L, M, N be distinct points in the plane such that A, B, C, D are the vertices of a square with sides AB, BC, CD, DA and L, M, N lie on the sides AB, CD, BC respectively. If M is the ...
6
votes
2answers
79 views

Proving a triangle equilateral given condition $al_a^2+bl_b^2+cl_c^2=9R\Delta$

$ABC$ is a triangle, with $l_a$, $l_b$, $l_c$ as angle bisectors, $R$ as circumradius and $\Delta$ as area, such that: $$al_a^2+bl_b^2+cl_c^2=9R\Delta$$ Is it true that $ABC$ is equilateral? I am ...