For questions about properties and applications of triangles

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2answers
72 views

Prove triangles formed by two midpoints and an altitude are congruent

Triangle ABC has altitude BH. M is the midpoint of AB, and N is the midpoint of CB. Prove triangle MBN is congruent to triangle MHN. Can we say that MN bisects BH? If so, why? If MN bisects BH (at ...
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1answer
69 views

Prove triangle made from two altitudes and midpoint is isosceles

In triangle ABC, AH and BK are altitudes. M is the midpoint of AB. Prove that triangle MHK is isosceles. All I can see is that the angles formed where the altitudes intersect are equal, and since ...
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2answers
70 views

In any triangle ABC, the expression (a + b + c) (a + b - c) (b + c - a) (c + a - b)$ is equal to

In any triangle ABC, give an equivalence to the expression $$(a + b + c) (a + b - c) (b + c - a) (c + a - b)$$ Can somebody help me? Note that ...
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4answers
197 views

Construction of a triangle

I need to construct a triangle with given information: $c = 6$, $h = 4$ and $\alpha - \beta = 30º$. I'll put approximate result for any clarification.
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1answer
42 views

Calculate PQ if AC = 20

I need to calculate PQ knowing that AC = 20. This is what I got so far: If I call the point between P and A, "M" and If I call the angle: $$\measuredangle{QPB} = y$$ Then: ...
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4answers
115 views

Ratio of Areas of Similar Triangles

First step, I can't find the height. How do you find the height?
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2answers
52 views

Find the value of $a$.

please help I'm lost on what numbers to add or what formula to use
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1answer
414 views

Fourier transform of a triangular pulse

So I've been practicing some fourier transform questions and stumbled on this one; To start off, i defined the fourier transform for this function by taking integral from -tau to 0 and 0 to tau as ...
2
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1answer
32 views

Do the medians (or other cevians) form all the triangles?

I want to know whether set of medians of all triangles, or some other class of cevians, can form the set of all the triangles? For example, in the case of altitudes, $(4,7,10)$ is an counterexample. ...
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1answer
28 views

Translate line vertically and calculate intersection on circle

Let's say I have a line extending from the center of the circle at a 45° angle. If I were to translate that line up 212.132 units, how would I calculate the intersection between the translated ...
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1answer
39 views

Geometry, Mensuration

If the diagonal BC passes through center of the circle, then the area of the shaded region in the given figure is \begin{align*} a)\quad &\dfrac{a^2}{2(3-\pi)}\\ b) \quad ...
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1answer
26 views

$S$, $I$, $O$ are circumcenter, incenter and orthocenter then $SO\ge IO \sqrt2$

Let $S$, $I$ and $O$ be the circumcenter, incenter and orthocenter of $\triangle ABC$ then prove that $SO\ge IO \sqrt2$, or equivalently $SO^2\ge 2IO^2$. I was able to derive an expression for $SO^2$ ...
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0answers
18 views

Complete Triangle Given 3 Parallel Planes and 2 Points

I have a problem where a point B connects to a point C at a known angle and distance. Both point B and C are on two separate parallel axis, GH and JK respectively. I need to find a third point, A, on ...
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2answers
37 views

isosceles and tight triangle

Hi, I was wondering if there is a way to find x with only knowing the length of isosceles triangle and no other piece of information.
2
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1answer
42 views

Sum of inradius of constructed triangle

Let $ABC$ be a triangle with inradius $r$ and circumradius $R$. Let $A′B′C′$ be the triangle for which $A′B′$ is the perpendicular to $OC$ through $C$ and so on. Let $r_1$ be the inradius of $A'BC$, ...
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2answers
82 views

Maximum Area of a Triangle when 1 Side, Perimeter Known

This is an example of a "quantitative comparison" question the GRE would test. Suppose the following information is known: one side of a triangle has length 12 the perimeter of the triangle is 40 ...
2
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2answers
51 views

How to find the area of an isosceles triangle without using trigonometry?

I have an isosceles triangle with equal sides $10$ unit, angle between them is $30^\circ$. I need to be confirmed that the area of this triangle can be found in any method other than using any kind ...
4
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4answers
87 views

Construction of an equilateral triangle from two equilateral triangles with a shared vertex

Problem Given that $\triangle ABC$ and $\triangle CDE$ are both equilateral triangles. Connect $AE$, $BE$ to get segments, take the midpoint of $BE$ as $O$, connect $AO$ and extend $AO$ to $F$ where ...
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3answers
55 views

For a triangle $ABC$, $a^2+b^2+c^2=8R^2$ then it is a right triangle?

$ABC$ is a triangle, $a^2+b^2+c^2=8R^2$ then how do we prove it is a right triangle?
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1answer
53 views

Triangle question, proving isoceles given trigometric conditions

$ABC$ is a triangle satisfying the following condition: $$\frac{\sin B}{\sin A}=\frac{\tan B+\cot C}{\tan A+\cot C}$$ How do I prove that $ABC$ is isoceles? I really have no idea.
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1answer
33 views

How prove that $|QA| < |QC|$ in triangle?

$ABC$ is a triangle with a right angle at $A$, and $|AB|$ > $|AC|$. The point $D$ is defined so that $BCD$ is equlateral and $AD$ intersects $BC$ at $P$. The point $Q$ is defined so that $QDP$ is ...
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2answers
41 views

How do I find a missing angle using a reciprocal trigonometric function?

I just attempted this as best as I could, but I'm not sure if I'm correct. Here's the work: $$\cot x =\frac{1}{2}$$ $$\frac{1}{\tan{x}} = \frac{1}{\frac{1}{2}}$$ $$\frac{1}{\tan^{-1}\cdot\tan x} = ...
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1answer
35 views

If $\frac1{HB}-\frac1{HA}=\cot C \cdot (\frac1{BC}-\frac1{AC})$, where $H$ is the orthocenter, then $ABC$ is isoceles?

If given that for a triangle $ABC$, with orthocenter $H$:$$\frac1{HB}-\frac1{HA}=\cot C \cdot (\frac1{BC}-\frac1{AC})$$ Then prove or disprove that $BC=AC$. How should I proceed with this?
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1answer
82 views

How show that $ABC$ is equilateral?

Let $D$, $E$ and $F$ be three points on sides $BC$,$AC$ and $AB$ of triangle $ABC$ such that lines $AD$, $BE$ and $CF$ concur at point $M$. If three trianles $MDB$, $MCE$ and $MAF$ have equal areas ...
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1answer
44 views

How prove that $AD>BE$ in triangle?

Let $D$ be a point on the side $BC$ of a triangle $ABC$ such that $AD>BC$ . The point $E$ on $CA$ is defined by the equation $\frac{AE}{EC}=\frac{BD}{AD-BC}$ .How prove that $AD>BE$?
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2answers
65 views

Geometrical proof for $PA+PB+PC\le3R$, where $P$ is the orthocenter and $R$ is the circumradius

$ABC$ is an acute angled triangle, where $P$ is the orthocenter, and $R$ is the circumradius. I want to show that $PA+PB+PC\le 3R$ geometrically, that is without using trigonometry. I have a trig ...
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1answer
70 views

Splitting a triangle to make two equal halves, find the length of the new line

Could someone please explain to me how I would find this out? I have a triangle and I need to find the length of the line that would split it down the middle so that the areas were even. A = 105 ...
2
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2answers
283 views

Area of Triangle when 2 Sides and No Angle Known

It is quite possible this question has no answer -- that is, the area cannot be determined from the information given. It's a question I've created myself as I study for the GRE. No trigonometry is ...
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2answers
92 views

Find side BC of a triangle given AB, AC, and a relation between $\angle A$ and $\angle B$

A question from my class: In triangle $ABC$, $3\angle A+2\angle B=180$ and $AB=10, AC=4$. So question is, what all can we comment on side $BC$. Can we find its exact length? I have a crude ...
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2answers
53 views

Find the angle between the sides 4 and 7 in a right triangle

I need to solve the $B$ corner What I've tried: $$\operatorname{sin} B=\frac47$$ $$B=\operatorname{arcsin}\frac47$$ $$B=34.85$$ But that's not the right answer, can anyone help me find what I did ...
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0answers
64 views

How to prove that three points are collinear [closed]

If H is the point within triangle ABC prove that the external bisector of the angles of AHB, BHC, CHA meet AB, BC, CA respectively at three collinear points. I don't have any idea how to solve this ...
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2answers
52 views

Proving the following inequality in a triangle

In a triangle the straight lines $AD$, $BE$, $CF$ are drawn through a point $P$ to meet $BC$, $CA$, $AB$ at $D$, $E$, $F$ respectively: Prove that $$\frac{PD}{AD} + \frac{PE}{BE}+\frac{PF}{CF}=1$$ ...
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1answer
23 views

Incentre of the triangle proving

A straight line is drawn through the incentre I of the triangle ABC perpendicular to AI meeting AB, AC in D and E respectively. Prove that BD.CE=ID^2
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4answers
78 views

How to find an angle (in degrees) in a right triangle, given its sides?

I need to find out a degree of an angle. Pretty simple, or so I thought. I remember doing a crap-ton of these in high-school, sadly the details did not remain. Anyway, let's take a look at this ...
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3answers
38 views

Basic question about angles

Why is the answer a)? Why can't it be d)? Why are the choices listed in this format, i.e., $(x \pm \theta^{\circ})$, and why is angle C $(x+30^{\circ})$ and not just $30^{\circ}$? Thanks.
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2answers
78 views

Simple proof of existence of hyperbolic triangles

I've studied the hyperbolic plane by analytically building up the hyperboloid model, the Klein—Beltrami disc, the Poincaré disc, and the half-plane model from scratch. Now I'd like to prove that, ...
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1answer
186 views

How to get the third point coordinates in isosceles triangle?

Isosceles triangle $ABC$ $AB = AC = d_1$ $BC = d_2$ $A = (x_1, y_1)$ $B = (x_2, y_2)$ $C = (x_3, y_3)$ $\angle BAC = \phi$ $\angle ABC =\angle ACB = \theta$ I want an equation for $x_3$ and $y_3$ ...
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1answer
82 views

Number of triangles formed by all chords between $n$ points on a circle

We have $n$ point on circumference of a circle. We draw all chords between this points. No three chords are concurrent. How many triangles exist that their apexes could be on circumference of ...
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1answer
58 views

problem about length of perpendicular chords

Question $AB$ is chord of circle $O$,points $D$ and $E$ are chosen on $AB$ in a way that $AD=BE$.prove two chords that are perpendicular to $AB$ and pass $D$ and $E$ points are equal.(prove $LK=MN$) ...
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1answer
28 views

Triangle Theorem relating the shortest and longest distance from any arbitrary point inside

I recall somewhere there was a relationship such that given a triangle T and a point A: if A is inside of T, then the sum of the longest distance from A to any point on a side of T, plus the shortest ...
2
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1answer
73 views

Trigonometric Substitution and the Triangle Inequality

I was reading the solution to this problem: If $x, y, z$ are real numbers and $x+y+z=xyz$ then $x(1 − y^2 )(1 − z^2 ) + y(1 − z^2 )(1 − x^2 ) + z(1 − x^2 )(1 − y^2 ) = 4xyz$ The solution is to ...
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1answer
33 views

Find points of triangle, one point, all sides and all angles known

Imagine the setup above; how can I calculate the points P1 and P2 if all angles, all sides A,B,C and point P3 are known?
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0answers
43 views

Trigonometry, find distance of arc movement

Imagine I have the setup as follows: I want to compute the height x in State 2, depending on how much the blue axis have moved. That is, the distance ...
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1answer
63 views

Circle with perpendicular chords

A blue circle is divided into $100$ arcs by $100$ red points such that the lengths of the arcs are the positive integers from $1$ to $100$ in an arbitrary order. Prove that there exists two ...
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1answer
94 views

Minimum Value of $x_1+x_2+x_3$

For an Acute Triangle $\Delta ABC$ $$\begin{align}x_n=2^{n-3}\left(\cos^nA+\cos^nB+\cos^nC\right)+\cos A\,\cos B\,\cos C\end{align}$$ Then find the least value of $$x_1+x_2+x_3$$ My Approach: I have ...
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1answer
104 views

Rotation matrix of triangle in 3D

How can I find out the rotation matrix of a right angle triangle defined by 3 points in 3D space (assuming the un-rotated triangle faces the x axis)
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0answers
35 views

Find angle of an arc in the circle using 3 coordinates

I want to find angle of semicircle. I have 3 coordinates (center_a,center_b) , (pivot_a,pivot_b) and (last_point_a, last_point_b). From triangle , i can find angle using equation using cosine ...
2
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1answer
47 views

Area of a triangle whose each side is less than 2 and greater than1.

What is the area of a triangle if each of its sides is greater than 1 and less than 2? My Try:Let a,b,c be the sides of triangle,then ...
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3answers
27 views

Locus Similar Triangle

ABC is a triangle and XY is variable straight line parallel to AC meeting BC and BA in X, Y respectively. If AX and CY meet at P find the locus of P.
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0answers
43 views

Similar triangle, Quick question (Thick Lens Formula)

http://www.panohelp.com/thinlensformula.html On the right hand side, f is defined as focus of the lens, i understand why the image distance is (f + fm). However i have spent an afternoon and could ...