For questions about properties and applications of triangles

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0
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3answers
27 views

Percentage change using differentials.

We're given the above triangle with sides $a$ and $b$ , and area $A$. $a$ is increased by $4$% and $b$ is decreased by $3$% , we need to approximate the percentage change in the area using ...
-2
votes
1answer
58 views

Prove that $\frac{r_1}{r-r_1}+\frac{r_2}{r-r_2}+\frac{r_3}{r-r_3}=\frac{r_1r_2r_3}{(r-r_1)(r-r_2)(r-r_3)}$ [on hold]

Let $D,E,F$ be the feet of the perpendiculars from the incenter $I$ to the sides $BC,CA$ and $AB$ respectively. If $r,r_1,r_2$ and $r_3$ are the inradius of the triangle $ABC$ and radii of the circles ...
0
votes
0answers
33 views

Show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$

If the internal bisectors of the angles of the triangle ABC make angles $\alpha,\beta,\gamma$ with sides $a,b,c$ respectively then show that $a\sin 2\alpha+b\sin 2\beta+c\sin 2\gamma=0$ I tried to ...
2
votes
2answers
37 views

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$

If the circumcenter of the triangle $ABC$ is on the incircle of the triangle,then prove that $\cos A+\cos B+\cos C=\sqrt2$ How should i attempt this question?I thought over it hard but could not ...
3
votes
4answers
43 views

$R$ is the midpoint of $MN$ and the points where $AC$ intersects $MD$ and $ND$ are $P$ and $Q$, respectively. Show that $PR=QR$.

In square $ABCD$, $M$ and $N$ are points on $AB$ and $BC$, respectively such that $\angle MDN=45°$. $R$ is the midpoint of $MN$ and the points where $AC$ intersects $MD$ and $ND$ are $P$ and $Q$, ...
2
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0answers
31 views

Geometry/Trigonometry Determine angle in a Triangle [duplicate]

Triangle ABC is isosceles with BC as base, AB=AC and Angle A=20 degrees. Points D and E lie on sides AB and AC respectively, such that D lies between A and B, E lies between A and C, angle BCD=50 ...
0
votes
1answer
322 views

A problem on triangle and its perpendicular bisectors.

I'm trying to solve the following problem : "In △ABC, coordinates of $B$ are $(−3, 3)$. Equation of the perpendicular bisector of side $AB$ is $2x + y − 7 = 0$. Equation of the perpendicular ...
-1
votes
0answers
37 views
+50

Prove that $\sin\theta_1.\sin\theta_2.\sin\theta_3=\frac{r^2_1}{16R^2}$

If $2\theta_1,2\theta_2,2\theta_3$ are the angles subtended by the circle escribed to the side $a$(opposite to vertex $A$) of a triangle at the centers of the inscribed triangle and the other two ...
1
vote
0answers
3 views

Right triangle and Sine function?

Given two angles and the hypotenuse of a right triangle, when trying to find the length of the side opposite the given angle, why and how does it's angle and supplementary angle yield the same answer? ...
1
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3answers
21 views
2
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1answer
358 views

Geometry - optimal 2D mesh between X expendable points

Say you have X points on a plane. If you connect two points, you form a line. Connecting three points forms a triangle. A line cannot cross a line, and a smaller triangle cannot be created inside a ...
6
votes
3answers
182 views

Let $M$ be an arbitrary point located inside the triangle $ABC$. Prove that $\cot\angle MAB + \cot\angle MBC + \cot\angle MCA \geq 3\sqrt{3}$

Let $M$ be an arbitrary point located inside the triangle $ABC$. Prove that $$\cot\measuredangle MAB + \cot\measuredangle MBC + \cot\measuredangle MCA \geq 3\sqrt{3}$$
0
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0answers
23 views

Double integral over a triangle

Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a smooth function (derivable, integrable over all of $\mathbb{R}^2$). Let $T$ be a triangle in $\mathbb{R}^2$, defined by its vertices : $A=(x_a,y_a)$, ...
1
vote
3answers
321 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 ...
0
votes
1answer
339 views

How can I transform a 3D triangle to xy plane

Suppose I am given a triangle ABC and its corresponding vertex coordinates in 3D. I want to transform ABC in such a way so that vertex A lies on global (0,0,0) coordinate, B lies on (dist, 0, 0) ...
1
vote
2answers
32 views

Need help proving this geometry problem.

My friend asked me one question yesterday.It is as follows. Let there be two triangles ABD and ACD.D is a point on base BC such that BD=CD(given).Also,clearly side AD is common.Now we know median ...
5
votes
3answers
175 views

Prove that $\angle AQP+\angle NAP=90^o$

Let a triangle $ABC$ be right at $A$, $AH$ be the altitude to the side $BC$. Let $M$ be an arbitrary point located in $AH$. Draw circle $B$ with the radius $BA$, circle $C$ with the radius $CA$. ...
0
votes
1answer
46 views

Is there enough information to answer this question?

My daughter got this question and I cannot solve it - or even give her direction. It appears there in not enough information. the number of equilateral triangles of side 1 into which an equilateral ...
0
votes
2answers
8 views

Angles and polygons. [closed]

The ratio of the interior angle to the exyerior angle of a regular polygon is 5:2. Find the number of sides of the polygon. Note the polygon may be three sided
18
votes
6answers
29k views

Finding out the area of a triangle if the coordinates of the three vertices are given

What is the simplest way to find out the area of a triangle if the coordinates of the three vertices are given in $x$-$y$ plane? One approach is to find the length of each side from the coordinates ...
0
votes
2answers
23 views

Right triangle trigonometry help?

I've got a right triangle where I know the slope of side $c$ based on the two points $(-150,200)$ and $(0,0)$. Also I know the length of side $a$. I was wondering based on these two known factors how ...
8
votes
1answer
247 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 ...
-2
votes
1answer
20 views

Relations involving the altitudes and orthocenter of a triangle [closed]

For acute $\triangle ABC$ with altitudes $AD$, $BE$, $CF$, orthocenter $H$, and area $S$, I have to prove that: $$AB^2 + HC^2 = BC^2 + HA^2 = AC^2 + HB^2 \tag{a}$$ $$AB \cdot HC + BC \cdot HA + ...
1
vote
1answer
37 views

Maximum number of equilateral triangles in a circle

I am stuck with a question. Given a circle with radius $x$ cm, what is the maximum number of equilateral triangles of side length 1 cm that can fit in the circle without overlapping or ...
0
votes
1answer
85 views

An integral expressing the resistance of a triangular region

Say there is an acute isosceles triangle (40,70,70 degrees) with a height of 75 units. I would like to take the surface integral over the function $f(w,l)=w/l$ where $w$ is the width and $l$ is the ...
0
votes
1answer
25 views

Dividing a Triangle by Connecting the Midpoints of its Sides

If $T$ is any triangle. Suppose we connect the midpoints of its sides forming four triangles. Does these four triangles have the same angles?
-1
votes
1answer
1k views

Find coordinates of vertex in right triangle

I have a right triangle with known points $A(x_1,y_1), B(x_2,y_2)$ and known cathetus $AC$ and $BA$. I need to find the coordinates of point $C$.
0
votes
1answer
30 views

If $AD=999$ and $PQ=200$, find the sum of the radii of those incircles.

Let $ABCD$ be a convex quadrilateral with $\angle DAB =\angle BDC=90° $. Let the incircles of $\Delta ABD $ and $ \Delta BCD $ touch $BD$ at $P$ and $Q$, respectively with $P$ between $Q$ and $B$. If ...
1
vote
3answers
992 views

How to find sum of 3 perpendiculars of a triangle?

Q. ABC is an equilateral triangle with side 10cm and P is a point inside the triangle, at a distance of 2cm from AB. If PD, PE and PF are perpendiculars to the three sides, find sum PD+ PF+PE. ...
0
votes
1answer
133 views

Finding the exact area of a trapizium using similar triangles

IN the trapezium ABCD, the diagonals intercept at M. Let AM= a, BM= b, Cm = c and DM = d, and let Angle AMB be $\theta$. a=6 b=4 c=3 d=2 AB=8 DC=4 $\cos(\theta) = -\frac{1}{4}$ and $\sin ...
1
vote
2answers
194 views

Trigonometry? Get the “half” of a triangle from hypotenuse and cathetus

I've only got the following parts of a triangle: Line A to B Line B to C And optionally the Line from A to C if needed? I'm trying to get the point X Now the problem is, i've got absolutly no ...
0
votes
2answers
16 views

Finding vector form of an angle bisector in a triangle

Find vector form of angle bisector, $\vec{BP}$, using $\vec{b}$ and $\vec{c}$. That's how far I've got. Please don't use $tb+ (1-t)b$, or similar since I don't know what that is. Just basic dot ...
3
votes
3answers
2k views

How is the hypotenuse the longest side of any right triangle?

I see that the hypotenuse of a right triangle is opposite the right angle, but how is it always the longest side? I also know that it connects to endpoints of other sides. Please help me out with ...
0
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0answers
37 views

Definition of triangle

If a polygon has 3 sides, but one side has zero length (or one angle is zero degree), is it still a triangle by definition of triangle? and how about if it has 2 sides, if not 3 sides are zero?
3
votes
0answers
45 views

“Natural” labeling of triangles

The angles of a triangle are (capital) $A,B,C$ and the lengths of the sides are (lower-case) $a,b,c$. At your mother's knee, you were taught that the side whose length is called (lower-case) $a$ ...
5
votes
5answers
426 views

What is the least number of (fixed) parameters I can ask for, when calculating area of a triangle of unknown type?

I need to calculate the area of a triangle, but I don't know, whether it is right angled, isoscele or equilateral. What parameters do I need to calculate the area of a triangle of unknown type?
2
votes
1answer
57 views

$ABCD$ is a square. $M$ and $N$ are points on $AB $ and $BC$, respectively such that $\angle MDN=45^\circ$…

$ABCD$ is a square. $M$ and $N$ are points on $AB $ and $BC$, respectively such that $\angle MDN=45^\circ$. $R$ is the midpoint of $MN$ and $P$ and $Q$ are the points where $AC$ is intersected by $DM$ ...
1
vote
1answer
34 views

Two cevians divide a triangle into 4 parts. Calculate the area of the 4th part, given the other 3.

Good day Here is the question: Connecting $AF$ and setting areas $\triangle ADF = x$ and $\triangle AFE = y$: $\frac {9+x}{12} =\frac y{15}$ $\frac{15+y}{12} =\frac x9$ from the ratios of the ...
1
vote
0answers
24 views

Line tangent to circle inside an isosceles triangle

If you take a circle enclosed inside an isosceles triangle, and then draw a line which is tangent to the circle and which intersects with the two equal sides, is that line parallel to the triangle's ...
7
votes
2answers
165 views

Find max of $S = x\sin^2\angle A + y\sin^2\angle B + z\sin^2\angle C$

Let $x$, $y$, $z$ are positive constants. $A$, $B$, $C$ are three angles of the triangle. Prove that $$S = x \sin^2 A + y \sin^2 B + z \sin^2 C \leq \dfrac{\left(yz+zx+xy\right)^2}{4xyz}$$ and find ...
-3
votes
2answers
28 views

how to find the number of integer coordinates in the interior of triangle

How to find the number of integer coordinates in the interior of the triangle with vertices(0,0) (0,21) (21,0).
1
vote
3answers
1k views

Triangle inscribed in circle, vertex at circle's center, solve for unknown angles.

$O$ is the center of the circle , $A$ and $B$ lie on the circle what are the possible values of $x$ and $y$ I found answers options , asked to mark one or more ...
3
votes
2answers
119 views

Generating Pythagorean Triples S.T. $b = a+1$

I am looking for a method to generate Pythagorean Triples $(a,b,c)$. There are many methods listed on Wikipedia but I have a unique constraint that I can't seem to integrate into any of the listed ...
1
vote
2answers
45 views

A Part of a semicircle between the two legs of a right angle triangle

In a right angled triangle, a semicircle is drawn such that its diameter lies on the hypotenuse and its center divides the hypotenuse into two segments of lengths 15 and 20.Find the length of the arc ...
1
vote
2answers
71 views

Find x in below diagram geometry

I am having difficulty in solving below question. Please help. Find x angle in below diagram I have drawn two parallel lines from D and E intersecting sides CB and CE respectively on F and G. look ...
-1
votes
0answers
31 views

Triangle inequality $ax ≥ br + cq$

I got stuck on this problem : Given a triangle (△ABC) of sides $a$, $b$ and $c$, let $O$ be a point inside △ABC. Let $D$, $E$ and $F$ be points on sides a, b and respectively c such that $OE ⊥ ...
4
votes
2answers
52 views

Angle chase:In $\Delta ABC, AB=AC $ and $\angle BAC=20°.$ If $CD$ is the median from $C$ to side $AB$, find $\angle ADC$.

In $\triangle ABC, AB=AC $ and $\angle BAC=20^\circ$ If $CD$ is the median from $C$ to side $AB$, find $\angle ADC$.
3
votes
1answer
43 views

Prove that the ratio of the areas of the triangles $A'B'C'$ and $ABC$ is $2\sin \frac{A}{2}\sin \frac{B}{2}\sin \frac{C}{2}$

If the bisectors of the angles of a triangle $ABC$ meet the opposite sides in $A',B',C'$,prove that the ratio of the areas of the triangles $A'B'C'$ and $ABC$ is $2\sin \frac{A}{2}\sin \frac{B}{2}\sin ...
3
votes
0answers
38 views

Area of $A'B'C'$ is to area of $ABC$ is $\frac{(m-n)^2}{m^2+mn+n^2}$

In the sides $BC,CA,AB$ are taken three points $A',B',C'$ such that $BA':A'C=CB':B'A=AC':C'B=m:n$.Prove that if $AA',BB',CC'$ are joined they will form by their intersections a triangle whose area is ...
2
votes
3answers
40 views

Triangle area inequalities

I've got stuck on this problem : Proof that for every triangle of sides $a$, $b$ and $c$ and area $S$, the following inequalities are true : $4S \le a^2 + b^2$ $4S \le b^2 + c^2$ ...