For questions about properties and applications of triangles

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0
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3answers
25 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
56 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
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0answers
27 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
34 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 ...
2
votes
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 ...
3
votes
4answers
40 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 $Q$, ...
1
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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
vote
3answers
21 views

Finding a coordinate over a right angle in a triangle where the other two coordinates are known

a B ------- C \ | \ | \ | c \ | b \ | \ | \| A Alright, this is a triangle I have, and ...
0
votes
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
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2answers
31 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 ...
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0answers
33 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 ...
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. [on hold]

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
-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
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 ...
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?
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 ...
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 ...
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
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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) $a$ ...
1
vote
1answer
33 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
23 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 ...
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 $DM$ ...
-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
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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
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 ⊥ ...
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 ...
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
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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 ...
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 ...
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 : $4S \le a^2 + b^2$ $4S \le b^2 + c^2$ ...
2
votes
4answers
45 views

Inequality between altitude and sides in triangle

Let $a,b,c$ be the side lengths and $h_a,h_b,h_c$ the altitudes each connect a vertex to the opposite side and are perpendicular to that side. Then we need to prove ...
0
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0answers
20 views

Proof of Menelaus using areas

I've tried to proof Menelaus' theorem using areas, but I've didn't figure out how. Some suggestions would be appreciated. Menelaus' Theorem states : Given a triangle ABC and a transversal ...
2
votes
3answers
40 views

Proving a ratio that has a relation with the Perpendicular bisectors and circumcircle

$ABC$ is a triangle, $D$ is a point on the side $BC$ of $\triangle ABC$, $R_b$ is circumradius of $\triangle ABD$ , and $R_c$ is the circumradius of $\triangle ACD$. Prove that $$ {Rb\over Rc} ...
1
vote
1answer
40 views

Prove by vector method that $p_1+p_2=p_3$

Let $ABC$ be an acute angled triangle whose incenter and centroid are respectively $I$ and $G$.$AI,BI$ and $CI$ cuts the sides of the triangle at $P,Q,R$ respectively.If $p_1,p_2$ and $p_3$ are the ...
0
votes
1answer
25 views

Using Tan to find the area of a triangle

I have come across a question that I can't seem to figure out. If tanA = 3/4, find the area of the given triangle without using a calculator The given triangle is an scalene triangle with a ...
4
votes
3answers
405 views

Find the longest side of the triangle.

The sides $a,b,c$ of a $\triangle ABC$ are in $GP$ whose common ratio is $\frac{2}{3}$ and the circumradius of the triangle is $6\sqrt{\frac{7}{209}}$.Find the longest side of the triangle. I used ...
0
votes
1answer
79 views

Geometry - angle bisector, circumcircle: SL olympiad

I tried this problem as much as I can, but I got nothing. This is a Sri Lankan mathematical olympiad problem. Let $P$,$Q$ be points on the sides $AB$ and $AC$, respectively, of a $\triangle ABC$ ...
8
votes
1answer
243 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 ...
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?
0
votes
2answers
22 views

How to scale a triangle such that the distance between original edges and new edges are equal?

This is very similar to this question: Coordinates of parallel triangle with a distance of 'd' between the parallel edges? That seems to provide the answer in 2D, but I am unsure how to ...
0
votes
1answer
36 views

Find triangle $ABC$ satisfies $1+2\sqrt{2}\sin\frac{B}{2}\sin\frac{C}{2}=\cos B+\cos C$

$\color{Red}{\texttt{Find all the triangle ABC}}$ whose angles satisfies $$2\left (1+tan^2\frac{C}{2} \right )\left [ cos^2\left (\frac{13\pi }{2}+\frac{B}{2} \right ...
-1
votes
2answers
32 views

Find area of an isosceles triangle with medians perpendicular to each other and base of length 4. [closed]

Hello guys can you help me with this problem.thanks in advance.
5
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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 $CA$. ...
2
votes
2answers
63 views

Triangle Area problem

I've been trying to solve the following: Let $ABC$ be a triangle with sides $a, b $ and $ c$, inradius $r$ and exradii $r_a, r_b$ and $r_c$. If $A'B'C'$ is another triangle with sides $\sqrt{a}, ...
0
votes
1answer
46 views

$\frac{1}{\Delta}\sqrt{abc(a+b+c)}$

If the triangle ABC has sides $a,b,c$ opposite to the vertices A,B,C respectively and $\Delta$ is the area.The expression $\frac{1}{\Delta}\sqrt{abc(a+b+c)}$ is $(A)\leq16\hspace{1 ...
0
votes
0answers
29 views

. Find the projection of the triangle on the coordinate planes.

Given the following, three vectors: a⃗ =3i−2j+5k b⃗ =i−6j+6k c⃗ =2i+3j−k Relative to cartesian coordinate systems with origin O. I calculated the sides to be 4.58,11.45 and 7.87. I also calculated ...
2
votes
1answer
50 views

Find all integer sided triangles whose area equals their perimeter.

Find all integer sided triangles whose area equals their perimeter? Taking sides to be $e,f,g$ we have, $$\sqrt{(e+f+g)(e+f-g)(e+g-f)(g+f-e)}=4(e+f+g)$$ how to proceed next ?
6
votes
3answers
180 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 $$\cot\measuredangle MAB + \cot\measuredangle MBC + \cot\measuredangle MCA \geq 3\sqrt{3}$$