For questions about properties and applications of triangles

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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: $$ ...
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1answer
144 views

Geometry: Construct a rectangle with area equal to a given triangle and with one side equal to a given segment.

So we have triangle, ABC, and line segment DE. We don't know anything about them other than ABC is a triangle of some sort and DE is a line segment. We're tasked with constructing a rectangle such ...
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2answers
41 views

inequalites of an acute triangle angles $ 180^{180}*a^b*b^c*c^a \le (a^2+b^2+c^2)^{180} $

If $a,b,c$ are an acute angle of triangle the prove that $ 180^{180}*a^b*b^c*c^a \le (a^2+b^2+c^2)^{180} $ No idea
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2answers
32 views

Calculating two points on two different circumference of circle

Given the center point[(x1, y1), (x2, y2)], the radius(r1, r2), how to calculate the coordinate of two points on the circumference of circle? I have drawn a picture, the two points marked as red in ...
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0answers
64 views

How to find an angle of a non-right angle triangle when given two sides and an area?

How would I go about finding an angles of a non-right angled triangle when given the area and two of its sides. For example: In the triangle $ABC$, $a = 5$, $b = 6$ and the area is $11~\text{cm}^2$. ...
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46 views

Efficient way to check whether triangles are similar

If we need to find if a triangle is isosceles, we can compare like a=b, b=c and a=c. But there are 3 comparisons. With $(a-b)(b-c)(a-c) = 0$, we can check it with one comparison. Usually for similar ...
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2answers
112 views

Prove a length of 6 in a triangle diagram.

A puzzle: Three equilateral triangles of size 3, 4, and 7 touch at a corner. The other corners of the size 4 triangle are 3 away from a 3 corner, and 7 away from a 7 corner. How far apart are the ...
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1answer
151 views

Finding the angle of elevation, three points on ground know angles of two of them.

This would be a pain to clearly write out, so I've made a picture of the exact set up: I need to find the angle of elevation from point C. It's supposed to be $75^\circ$. I've tried using $\sin$, ...
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2answers
60 views

I need to prove that this line is a tangent to the circle

The problem is this: Given two different points, $A$ and $B$, take the midpoint between them ($O$) draw the circumference $\Gamma (O,OA)$ Take any point $C$ on $AB$ and draw a line $t$ perpendicular ...
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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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How one can show Gerretsen's inequality?

I read from http://rgmia.org/papers/v6n3/wsh.pdf the following: A triangle with semiperimeter $s$, circumradius $R$ and inradius $r$ satisfies $$16Rr-5r^2\leq s^2\leq 4R^2+4Rr+3r^2.$$ How can I prove ...
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2answers
49 views

Trignometric inequality

if $\alpha, \beta, \gamma$ are angles of a triangle. prove that $\csc(\frac\alpha2)+\csc(\frac \beta2)+\csc(\frac \gamma2) \ge 6$. I started from $\alpha + \beta + \gamma = 180^{\circ}$ and then I ...
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3answers
56 views

Calculate the length of the sides of a triangle from the area and the angle

I need to find the length of the sides of a triangle. I have an angle and the area of the triangle. I have the answer but I don't know how to figure it out so it doesn't help. The area of the ...
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1answer
46 views

Finding Triangle's Angle

Find the value of $x$ if the area is $42\;\mathrm{cm}^2$. The triangle is not right angled or an isosceles. $X$ is the angle in the bottom left corner of the triangle that I need to find. The left ...
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63 views

Let $a,b,c$ be the lenghts of the sides of a triangle. Suppose that $ab+bc+ca=1$. Show that $(a+1)(b+1)(c+1)<4$

Let $a,b,c$ be the lenghts of the sides of a triangle. Suppose that $ab+bc+ca=1$. Show that $(a+1)(b+1)(c+1)<4$. My attempt: I tried multiplying the whole thing but that didn't help at all. ...
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2answers
35 views

Diagonal line through rectangle always creates two congruent triangles (?)

1) Is it true that the diagonal line through a rectangle always creates to congruent triangles? 2) If a quadrilateral has two right angles that are opposite (is this the right word to use), as ...
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4answers
85 views

In a triangle, prove that $a\cos A+b\cos B+c\cos C=\frac{8\Delta^2}{abc}$

I have to prove that for a triangle, $$a\cos A+b\cos B+c\cos C=\frac{8\Delta^2}{abc}$$ where $a,b,c$ are the lengths of the sides opposite to the angles A,B,C respectively. I followed the following ...
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0answers
18 views

Length of a right triangle's hypoteneuse projected onto a sphere

Please forgive me if this is the wrong kind of question, but I need someone to verify or refute my work. One leg of a triangle has length, $b$ (base), resulting from angle theta swept out by a ray ...
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5answers
127 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 ...
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2answers
30 views

How to find the remaining segment of this triangle without the Law of Cosines?

I am given a triangle $T$ with vertices $A, B,$ and $C$ and the the following info about $T:$ $\overline{AC} = 6$ Km $\overline{BC} = 9$ Km The angle formed by these two segments is $120^{\circ}$ ...
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1answer
38 views

Area of an isosceles triangle where the tangents of some angles are in geometric progression

In $\triangle ABC$, $AB=BC$ and $\overline{BD}$ is an altitude. Point $E$ is on the extension of $\overline{AC}$ such that $\overline{BE}=10$. The values of $\tan CBE$, $\tan DBE$, and $\tan ABE$ form ...
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Solutions of triangles - proof

Question: For a triangle ABC, prove that: $$r_1 + r_2 + r_3 = r + 4R$$ Where $r_1,r_2,r_3$ represent the radius of the ex-circles opposite to angle A, B, and C respectively. $r$ represents the ...
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1answer
47 views

How to solve a triangle knowing just two legs?

Given a right triangle, if I know two of its legs are $a = 3,5$ and $b = 5,5$, is it possible to solve the triangle with this info? This is, can I determine all of its sides and all of its angles with ...
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0answers
31 views

Finding the coordinates of the third point in a triangle using simultaneous equations

I have 3 circles A,B & C that touch each other at tangents. The centre points of these 3 circles are to be joined to create a triangle. I know the coordinates of 2 of the circles centre points ...
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1answer
70 views

Triangles in geometry please help [closed]

find the length of the altitude to the hypotenuse of a right triangle whose sides have lengths 6.8 and 10 smaller part 3.8
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126 views

Prove that the circumcenter of $\triangle PIQ$ is on the hypotenuse $AC$.

In right angled $\triangle ABC$ with $\angle B=90 ^{\circ}$, $BD$ is an altitude on $AC$. $P,Q,I$ are the incenters of $\triangle ABD,\triangle CBD$ and $\triangle ABC$ respectively. Prove that the ...
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3answers
392 views

Find the length of the chord given that the circle's diameter and the subtended angle

A chord of a circle subtends an angle of 89 degrees at its centre. Find the length of the chord given that the circle's diameter is 11.4 cm. The problem I have here is that I can't visualise this ...
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1answer
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Geometry: Perpendicular tangent

I came up with this but I have not been able to solve it. I would really appreciate any help. Let $ABC$ be a triangle and let $\omega$ be its circumcircle. Produce the internal angle bisector of ...
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1answer
47 views

Proving that $KL$ bisects $AJ$ in a triangle?

Let $ABC$ be an acute triangle, and its incircle touch the sides $AB$ and $AC$ at $K$ and $L$. Let $J$ be the incenter of $\triangle BCD$, where $D$ is a point on $AC$ such that $BD=AB$. Prove that ...
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1answer
36 views

triangle circle inside it, heights prove exercise

Point $O$ lie inside $ABC$ triangle. Points $A1,B1,C1$ are projections of $O$ on heights led from $A,B,C$ Prove that if $AA1=BB1=CC1$ then $AA1=2r$, where $r$ is radius of circle inscribed in $ABC$ ...
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1answer
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Construction of a triangle given some special points ($O,H,I$)

I'm a newbie in this site. I tried to search if this question was already answered but I'm not sure on how to do it. The problem is: given three distincts points $O,H,I$ namely the circumcenter, the ...
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Trisecting the sides of a triangle.

Consider the hexagon formed by the six points which trisect the sides of a triangle(two on each side). Is is true that when we connect opposite points in this hexagon, the lines intersect at a single ...
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3answers
44 views

Find the area of the triangle under certain preconditions

With vertices $(0, 0)$, $(b, a)$, $(x, y)$, prove the area of this triangle is $\frac{|by - ax|}{2}$. We know area of a triangle = $\frac{rh}{2}$. ($r$ is the base.) Well, we have $r =$ the ...
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1answer
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Finding third vertexes of any triangle where 2 vertex known and all sides length known

I am working with a CAD engine in the head but i working on code only. I have a rectangular tube that need to be put at an angle. I so have the diagonal of the tube where it has to start and stop but ...
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1answer
76 views

Calculationg the angle of a triangle

I am trying to find a specified angle of a triangle. In triangle $ABC$, $\angle A = 20^\circ$. $D$ and $E$ are points on $AB$ and $AC$, where $AB=AC$. $\angle EBC = 50^\circ$ and $\angle DCB = ...
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0answers
42 views

Find the angle in a triangle [duplicate]

Find the angle $a$: I came up with 20 degrees but not sure. Can somebody help here.
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1answer
34 views

Relationships in a triangle

Here is the question, I can''t figure out how to explain this algebraically.
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1answer
171 views

Z coordinates of 3rd point (vertex) of a right triangle given all data in 3D

this is my first post.. I hope this good I have 1 triangle in space (3D)... and I know all data except the coordinates of 3er point(vertex)... for example this: then: ...
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1answer
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Find α (Triangles) [closed]

Find $\alpha$ if $A = 4\alpha$. Can someone explain to me how to do this?
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1answer
92 views

Prove that the intersection of $BM$ and $CN$ is on the circumcircle of triangle $ABC.$

Let $P$ and $Q$ be on segment $BC$ of an acute triangle $ABC$ such that $\angle PAB$ = $\angle BCA$ and $\angle CAQ = \angle ABC$.Let $M$ and $N$ be the points on $AP$ and $AQ$, respectively, such ...
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2answers
160 views

Geometry question about centroid [closed]

How do you solve this geometry question? In triangle $ABC$ the centroid is $G$ and $D$ is themidpoint of $CA$. The line through $G$ parallel to $BC$ meets $AB$ at $E$. Prove that $\angle ...
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1answer
64 views

Triangle Area Ratio Theorem Problems?

Having a hell of a lot of issues with these problems, supposed to be on the topic of triangle area ratio theorem (ratio area of triangles = ratio of triangles' heights x ratio of triangles' bases.) ...
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1answer
60 views

Proof of a set of triangles and unit squares

Suppose that there is $S$, a finite set of unit squares. So, $S$ is chosen from a larger grid of unit squares. The unit squares of $S$ are tiled with isoceles right triangles. Each of these triangles ...
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3answers
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Proving that $BI$, $AE$ and $CF$ are concurrent?

Let $ABC$ be a triangle, and $BD$ be the angle bisector of $\angle B$. Let $DF$ and $DE$ be altitudes of $\triangle ADB$ and $\triangle CDB$ respectively, and $BI$ is an altitude of $\triangle ...
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1answer
108 views

Inequalites of triangle side with $abc = 1$

Let $a,b,c$ be the sides of a triangle with $abc=1$. Prove that $$ \frac{\sqrt{b+c−a}}{a} + \frac{\sqrt{c+a-b}}{b} + \frac{\sqrt{a+b−c}}{c} \ge a+b+c $$
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1answer
37 views

How prove that $AB>AC$ in triangle $ABC$?

Point $D$ is chosen inside $\triangle ABC$, and point $E$ on segment $BD$ such that $BD=CE$. Suppose $\angle ABD=\angle ECD=10^{\circ}$, $\angle BAD=40^{\circ}$, and $\angle CED=60^{\circ}$.How prove ...
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1answer
68 views

Proving $B$, $C$, $D'$ and $E'$ to be concyclic iff $AB+AC=3BC$?

Let $ABC$ be a triangle with incenter $I$. The incircle of $ABC$ touches $AC$ at $D$ and $AB$ at $E$. Let $DD'$ and $EE'$ be the diameters of the incircle. Prove that $B$, $C$, $D'$ and $E'$ are ...
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1answer
27 views

Triangle Inequality with Vectors

If the magnitudes of vectors $\mathbf{a}$ and $\mathbf{b}$ are $5$ and $12$, respectively, then the magnitude of vector $(\mathbf{b-a})$ could NOT be (A) 5 (B) 7 (C) 10 (D) 12 (E) 17 The triangle ...
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1answer
57 views

Triangle $ABC$ and equilateral triangles $ABC'$, $BCA'$ and $ACB'$.

We consider a triangle $ABC$ whose angles are less then $120°$ and construct the equilateral triangles $ABC'$, $BCA'$ and $ACB'$, exterior to $ABC$. $I$ denotes the intersection of $(AA')$ and ...
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3answers
55 views

Given an equilateral triangle, show that $MA + MC = MB$.

I have to solve the following problem: Consider an equilateral triangle $ABC$ and $\mathcal{C}$ its circumscribed circle. Let $M$ be a point located on the arc of the circle defined by $[AC]$ which ...