4
votes
1answer
42 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 ...
9
votes
5answers
262 views

Tangent and angle bisectors

The tangent to the incircle of a triangle ABC is reflected about the external angle bisectors. Show that the triangle formed by the resulting 3 lines is congruent to ABC .
4
votes
3answers
167 views

Smallest square containing a given triangle

Given a triangle $T$, how can I calculate the smallest square that contains $T$? Using GeoGebra, I implemented a heuristic that seems to work well in practice. The problem is, I have no proof that it ...
3
votes
1answer
79 views

Circumcircle of an isosceles triangle and length relation

I was asked to prove the following problem. Consider the following diagram where a triangle $ABC$ lies inside its circumcircle, $D$ is the point where the angle bisector $\alpha$ of $B$ intersects ...
3
votes
2answers
227 views

What is the history of the use of the term “scalene triangle”?

A "scalene triangle" is a triangle with three unequal sides. As far as I can tell, this term is not in much use in serious mathematics — in fact, before I became a high school math teacher, I'd ...
0
votes
3answers
90 views

Perpendicular lines inside and outside a circle

No trigonometry allowed. Let $\Delta ABC$ be inscribed inside a circle.Let $P$ be a point on the circle.Let $PD$ and $PE$ be perpendiculars on on $BC$ and $AC$ respectively.Let $DE$ when extended ...
2
votes
1answer
47 views

Solution for the value of an angle of a triangle ABC

Find value of angle m< DBC Where $$BD=DC=AC$$ $$2(m\langle BAC)=14(m\langle ABD)=7(m\langle BCD)$$ I tried hard but im out of ideas now, I know the answer is 20 but I want to know how, thanks ...
0
votes
1answer
40 views

Determine missing angle in polygon

I'm trying to figure out this question: Determine the measure of angle a I'm guessing $a=96\unicode{0186}$ using the following work: $$a = 180 - 84 = 96 $$ ...
2
votes
1answer
90 views

Area of triangle inside triangle

In triangle $ABC$ we choose 3 points $D,E,F$, such that $\overline{AD} = \frac 13 \overline{AB}, \overline{BE} = \frac 13 \overline{BC}, \overline{CF} = \frac 13 \overline{CA}$. Draw segments ...
2
votes
2answers
102 views

How do I prove that $CP > \frac 1 2 (AC+BC-AB)$? [closed]

Given is the triangle $ABC$ with point $P$ on side $AB$. How do I prove that $$CP > \frac 1 2 (AC+BC-AB)?$$
4
votes
0answers
183 views

Area of a equilateral triangle given distances of a point in the triangle from the vertices [closed]

A point $D$ inside an equilateral triangle $PQR$. $D$ is located at a distance of $3$cm, $4$cm and $5$cm respectively from $P$, $Q$ and $R$. What is the area of the triangle $PQR$?
0
votes
0answers
60 views

Ratios in a rhombus

NOTE: I am NOT looking for a full answer,just a hint. Last problem on this question. BdMO 2013 Chittagong: Let $ABCD$ be a rhombus.Let $G$ be a point outside the rhombus such that GE is ...
1
vote
1answer
58 views

Angle bisector in a triangle

For the angle bisector $I_a$ in a triangle $ABC$ it holds $$I_a^2 = \frac{bc}{(b+c)^2}[(b+c)^2 - a^2]$$ If $I$ is the incenter, I wonder if there exist similar formula for the part $AI^2$.
1
vote
2answers
138 views

Find value of the angle x

Find the value of the angle x. Plus : Someone could recommend me some good book about this subject ?
3
votes
1answer
57 views

Formula in a triangle

Let $H$ be the orthocenter in a triangle with sides $a, b, c$. Is it true that $$a^2 + HA^2 = 4R^2$$ where $R$ is the circumradius?
0
votes
1answer
27 views

Figuring out the side of a triangle

I'm having trouble on this problem I don't know how to set it up. I know XO=2 and OB=6. I'd appreciate any hints.
0
votes
0answers
47 views

Triangle inscribed insemicircle area-ratio question

My approach: $m<A= 60$ degrees and $m<C=30$. This creats a 30, 60, 90 triangle with ratios $$1:\sqrt3:2$$ After getting the ratio's of the areas, I obtain $$\frac{b*h}{\pi r^2}=\frac{1*\sqrt ...
3
votes
1answer
116 views

What is the shape of the region where a special point can exist?

For an equilateral triangle $ABC$ which has edge-length $1$, let $D,E,F$ be a point of contact of the inscribed circle and an edge $BC, CA, AB$ respectively. Then, let us call the region surrounded ...
0
votes
1answer
51 views

Length of hypotenuse

Let a circle centered at $O$ have radius $OA=10$. Let OB be perpendicular on OA.Let G and E be points respectively on on OB and OA.Let F be a point on the circumference such that GFEO is a ...
0
votes
1answer
43 views

$3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle

I am completely lost in this one $3x+3y-1,4x^2+y-5,4x+2y$ are sides of an equilateral triangle, its area is closest to the which integer?
0
votes
2answers
44 views

How does this proof of law of sines determine equal angles?

I was reading over this proof of the law of sines and they say that $\angle CAB = \angle DOB$ because of "basic geometry". I do not get it though, how can you say that the angles are equal?
3
votes
1answer
98 views

Inequality in triangle

Let $ABC$ be a triangle and $M$ a point on side $BC$. Denote $\alpha=\angle BAM$, $\beta=\angle CAM$. Is the following inequality true? $$\sin \alpha \cdot (AM-AC)+\sin \beta \cdot (AM-AB) \leq 0.$$
3
votes
1answer
68 views

About the 'minimum triangle' which includes a convex bounded closed set

Question : Is the following true? "Letting $K$ be a convex bounded closed set on a plane, then there exists a triangle $M$, which includes $K$, such that $|M|\le 2|K|$. Here, $|M|,|K|$ is the area of ...
0
votes
1answer
62 views

About the area of integer-edge-length triangles

Let $a,b,c$ be three edge lengths of a triangle whose area is $S$. Then, here is my question. Question : Supposing that $a,b,c$ are natural numbers, then does there exists $(a,b,c)$ such that ...
2
votes
1answer
127 views

Lines $ MF, DE, QR$ in a triangle intersect at one point

In a triangle ABC, a circle is inscribed with center in $I$. The inscribed circle touches sides $BC,CA,AB$ in $D,E,F$ respectively. Join the point $C$ and $F$, $B$ and $E$. Let $Q$ and $R$ be the ...
15
votes
2answers
377 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 $ABC$ ...
5
votes
2answers
127 views

Find The range of $r/R$.

Given a triangle $ABC$ with angle $A=90^{\circ}$. Let $M$ be the midpoint of $BC$. If the inradii of the triangles $ABM$ and $ACM$ are $r$ and$\ R$ respectively, then find the range of $\dfrac rR$ .
4
votes
1answer
195 views

Could someone explain this animated gif to me in mathematical terms?

I understand that the area of the two squares around the right triangle are the total area of the one that is the hypotenuse. Is this just a proof for the Pythagorean theorem or is there some other ...
0
votes
1answer
201 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 ...
0
votes
1answer
21 views

How do I find out the coordinates, interpolating across an angled line?

Suppose I know the coordinates of $A$ and $B$. The angle $X$ does not mean the total angle between the red lines, but rather how far along the angle that the purple line is. What is the easiest way ...
1
vote
1answer
117 views

Triangle inequality for an obtuse triangle

$\alpha < 45^\circ$, how to show that 1) $|AB+AC|>|DB+DC|$? 2) $|AB+AC|>|DB+DC+DA|$?
1
vote
2answers
187 views

Does this proof work to prove that the greatest area of a triangle inside a circle is when the triangle is equilateral?

Does this proof work to prove that the greatest area of a triangle inside a circle is when the triangle is equilateral? I gather it doesn't because most of the proofs I've seen use derivatives etc. If ...
7
votes
1answer
674 views

Sum of distances from triangle vertices to interior point is less than perimeter?

Let $M$ be a point in the interior of triangle $ABC$ in the plane. Prove $AM+BM+CM<AB+BC+CA$. The above question was posed to someone I know who is taking high-school Euclidean geometry. I'm ...
3
votes
4answers
200 views

Maximum surface inside a triangle

If I have a triangle with sides of length a, b, c and I have a rope of length L, what is the maximum surface of a boundary I can form with that rope that is entirely inside the triangle. Normally, ...
5
votes
1answer
210 views

Sum of medians of a triangle

I'm very confused because I don't know how I can prove that the sum of the medians of a triangle is equal to the vector zero. Can someone give me a tip or something? Thanks! (And sorry if this ...
1
vote
2answers
208 views

A question on Trigonometry (bisector)

If two bisector of a triangular is equal, then it is Isosceles triangular.
6
votes
2answers
138 views

Concurrency of A'L, B'M, C'N.

Need some help with the following problem. Problem: In $\triangle ABC$ the midpoints of $BC$, $AC$, $AB$ are $L, M,$ and $N$ respectively, and the points on the circumcircle opposite to $A, B,$ and ...
1
vote
1answer
249 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 ...
6
votes
3answers
157 views

What characteristic of the triangle leads the the existence of the orthocenter

We all know that all three altitudes of a triangle meets in the orthocenter of the triangle. It's a quite classical problem and is proven. However, what I really wanna know is what characteristic of ...
1
vote
2answers
289 views

Triangle in hexagon

In a regular hexagon ABCDEF is the midpoint (G)of the sides FE and S intersection of lines AC and GB. (a) What is the relationship shared point of straight ...
2
votes
2answers
166 views

Proving that $|CA|+|CB|=2|AB|$ in a general $ABC$ triangle

How in this situation (presented in image) can I prove that $|CA|+|CB|=2|AB|$?
2
votes
1answer
249 views

Prove that the Simson line of $P$ bisects the segment $HP$ from the orthocentre $H$ to $P$

Let $ABC$ be a triangle with orthocentre $H$ and circumcircle $\odot(ABC)$. Suppose $P\in\odot(ABC)$. Let $\gamma$ be Simson's line of $P$ wrt $ABC$. Prove that $\gamma$ bisects $PH$.
2
votes
3answers
158 views

Sides of triangle and an altitude

Let $a$, $b$, $c$ be the lengths of the sides of a triangle. Let $h$ be the altitude drawn on the side of length $a$ Then is $a^2 + 4h^2 - (b+c)^2$ always negative ?
3
votes
2answers
316 views

Similarity involving Miquel's Theorem

Let $\Delta ABC$ be a triangle. If we place points $D,\ E,\ F$ arbitrarily on the sides $\overline{AB},\ \overline{BC}$ and $\overline{CA}$ respectively, then the circumcircles of the triangles ...
6
votes
3answers
906 views

Why is the inradius of any triangle at most half its circumradius?

Is there any geometrically simple reason why the inradius of a triangle should be at most half its circumradius? I end up wanting the fact for this answer. I know of two proofs of this fact. Proof ...
1
vote
1answer
624 views

Pythagorean theorem

We can make a square into four equal squares. Fine, if we want to make into five.. Then there is a problem. Please discuss, How to make five squares from a single square by using a Pythagorean ...
9
votes
1answer
656 views

Is there a value for $\pi$ that relates to triangles?

So I heard that if one inscribes the largest circle that can fit into a equilateral triangle, then divides the perimeter of the triangle by the diameter of the inscribed circle, it gives a value which ...
-1
votes
4answers
485 views

Why are there isosceles triangles? [closed]

Why are isosceles triangles called that — or called anything? Why is their class given a name? Why did they find their way into the Elements and every single elementary geometry text and course ...
2
votes
0answers
352 views

Euler's Line of a medial triangle

I have the following problem with a comment below on the steps that I took so far. Here is the example: Let triangle ABC be any triangle. The midpoints of the sides in Triangle ABC are labeled $A', ...
2
votes
2answers
437 views

Prove there exists a triangle without using Euclidean Parallel Postulate

Let $a$ and $b$ be real numbers where $0 < a< b<180$. Let $A$, $B$, $D$ be points so $A$-$B$-$D$. Part 1: Prove there exists a triangle $ABC$ where measure of angle $CAB$ is $a$ and measure ...