Tagged Questions

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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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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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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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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 ...
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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 .
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Get the angle in a circle using center, radius and one point in a circle.

There is a circle and i know Point1 this is fixed and i know another point Point2 which can be anywhere in the circle. and i want to know the angle which is made at center. Thanks Your help will be ...
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Circle theorem/triange angle question

I am doing practise papers and there is one question I cannot understand even with the mark scheme. I have added the pictures below: Question (with added annotations): Mark scheme: The question ...
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Maximal area covered by two triangles in unit circle

What is the maximal area covered by two triangles in a unit circle? There are no restrictions other than that. They can overlap, touch the circle, not touch the circle etc. So far I have shown In ...
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Find Coordinates on a track

Charlie and Alexandra are running around a circular track with radius 60 meters. Charlie started at the westernmost point of the track, and, at the same time, Alexandra started at the northernmost ...
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Finding coordinates on a circle

So this problem I am have difficulty with. I think where I am going wrong is how to calculate the initial theta. Do I just use pi/2 because in the pictures it show to angle theta off the 90 degree ...
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Complex Number and Geometry

Given $A(3+4i)$, $B(-4+3i)$ and $C(4+3i)$ be the vertices of a triangle $ABC$ which is inscribed in a circle $S=0$. Let $AD, BE, CF$ be altitudes through $A, B, C$ which meet the circle S=0 at ...
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Crazy rectangles, semi-circles, and circles!

Problem is to find the ratio of the area of the circle to that of the semi-circle. Note that points $F$ and $E$ weren't given in the original diagram, and that the circle at the top-right ...
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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 ...
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Formula to calculate a side of triangle with given angle

I have triangle like in the picture. The known angles: α (total angle of the I-J-K2 triangle) b (total angle of the I-P2-K2 and I-P1-K2 triangles) The known 3D points with X,Y,Z-coordinates: ...
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Relationship between circles touching incircle

I am trying to derive a relation between radius of those outer circles and radius of the incircle. Those outer circles are tangent to the incircle and respective sides. I have tried and failed ...
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Finding circumcentre

Tangents are draw from $P(2,3)$ to $x^2+y^2=4$ meeting at $Q,R$ on circle. Parallelogram $PQSR$ is completed. Find the circumcentre of triangle $QSR$. My attempt: Clearly, the parallelogram is a ...
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Computing distance in circle

It seems to me as pretty simple, but I just can't get hold of it: I am trying to compute fn(x, r). Thanks.
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Incenter of Triangle in 3D

I'm trying to figure out how to find the incenter of a triangle with (x, y, z) coordinates for the verteces. I can find the lengths of the sides and the radius of the incircle from that, so I've ...
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Three sides of a $\triangle$ are known. If a circle with it's center on base of $\triangle$ touches the other two sides , find the radius of circle.

In $\triangle ABC$, $AB = 10, AC = 12$ and $BC = 18$. A circle is drawn such that its center is on side $BC$ and it touches lines $AC$ and $AB$. Find the radius of the circle. By pythagoras ...
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How to prove that PH is containing midpoint of side MN from this circle and triangle problem?

Given: triangle ABC is acute triangle. M and N are midpoints of AB and BC respectively, while BH is altitude of triangle ABC. Circles AHN and CHM meet at point P. (P is not same with H) How to ...
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How to prove that the angle between two sides of that triangle is less than 60 degree?

The product of two sides of triangle is equal to 8*(R*r) where R is circumradius of this triangle, and r is inradius of this triangle. How to prove that the angle between two sides of that triangle ...
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Three circles with two common points

Let $ABC$ be a triangle of any type and $A_1,B_1,C_1$ the feet of the heights. Denote $M,N,P$ the orthogonal projections of the point $A$ onto the lines $B_1C_1,C_1A_1$ and $A_1B_1$. The circes ...
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Length of median extended to the circumcircle

A triangle has side length $13,14,15$, and its circumcircle is constructed. The median is then drawn with its base having a length of $14$, and is extended to the circle. Find its length.
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Sine defined for a triangle inscribed in a circle with a diameter of one

Let a circle be drawn with a diameter of one (and thus a radius of one half). Then let a triangle with vertices A, B, and C be inscribed in the circle (i.e. points A, B, and C are arbitrary points on ...
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Given 3 points and there distances from eachother find a fourth point equidistant to the 3.

This question can also be asked: given a triangle, and its dimensions, whose vertices lie on the edge of a circle find the radius of the circle. I am not actually sure if there is enough information ...
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How $\pi$, $3.1415…$ and $180^o$ are adaptive together?!

I planed following to compute the circle's circumference. The circle's circumference finally can computable from: $$\lim_{\alpha\to0}{\frac{360^o}\alpha d} = 2\pi r$$ I don't want to follow above ...
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Question on inscribed equilateral triangle

Question: $ABC$ and $ODE$ are equilateral triangle with $BC || DE$. If $O$ is the center of the circle, then find the ratio $AQ:QC$ So, my thought on this is that, since we are not given the ...