Let segment gh be a chord of a circle $\omega $ which is not a diameter, and let n be a fixed point on gh. For which point b on arc gh is the length n minimized?

Luciano Webster
2022-07-17
Answered

Let segment gh be a chord of a circle $\omega $ which is not a diameter, and let n be a fixed point on gh. For which point b on arc gh is the length n minimized?

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asked 2022-06-11

I found this in a textbook without a solution and I wasnt able to solve it myself.

Let ABCD be a tetrahedron with all faces acute. Let E be the mid point of the longer arc AB on a circle ABD. Let F be the mid point of the longer arc BC on a circle BCD. Let G be the mid point of the longer arc AC on a circle ACD.

Show that points D,E,F,G lie on a circle.

My approach to that was to try to show these point were co-planear. Since they all lie on one sphere (the one with inscribed tetrahedron ABCD) that would solve the problem. Needles to say I failed at that.

Let ABCD be a tetrahedron with all faces acute. Let E be the mid point of the longer arc AB on a circle ABD. Let F be the mid point of the longer arc BC on a circle BCD. Let G be the mid point of the longer arc AC on a circle ACD.

Show that points D,E,F,G lie on a circle.

My approach to that was to try to show these point were co-planear. Since they all lie on one sphere (the one with inscribed tetrahedron ABCD) that would solve the problem. Needles to say I failed at that.

asked 2022-06-02

I am given three co-ordinate points of a circle O(Ox,Oy) as a center. Then two Points other points as A(Ax,Ay) & B(Bx,By). Now I have to find the arc length of that circle. Can you please help. Thanks in advance.

asked 2022-06-24

A circle has diameter AD of length 400.

B and C are points on the same arc of AD such that |AB|=|BC|=60.

What is the length |CD|?

B and C are points on the same arc of AD such that |AB|=|BC|=60.

What is the length |CD|?

asked 2022-04-30

If the arc length and chord length between two points (two points on a circle that constitute a minor arc ) in a circle are known , find radius of the circle?

asked 2022-06-22

$arg(\frac{z-a}{z-b})=c$

My understanding is as follows. The angle c between the lines za and zb is constant. za and zb meet at z and the angle between the lines perpendicular to za and zb is 2c, which is only the case if z lies on a circle passing through z, a and b with centre where the lines perpendicular to za and zb intersect. So the locus is an arc of a circle through a and b except a and b (because $arg(\frac{z-a}{z-b})$ is undefined there). Is this reasoning correct?

Is there another way the statement could this be expressed (e.g. in terms of moduli)? Also, what is an efficient way of finding the centre of the circle?

My understanding is as follows. The angle c between the lines za and zb is constant. za and zb meet at z and the angle between the lines perpendicular to za and zb is 2c, which is only the case if z lies on a circle passing through z, a and b with centre where the lines perpendicular to za and zb intersect. So the locus is an arc of a circle through a and b except a and b (because $arg(\frac{z-a}{z-b})$ is undefined there). Is this reasoning correct?

Is there another way the statement could this be expressed (e.g. in terms of moduli)? Also, what is an efficient way of finding the centre of the circle?

asked 2022-08-17

Perhaps a rather elementary question, but I simply couldn't figure out the calculations on this one. Say one takes a circle centeblack at the origin with radius $R$. He or she then proceeds to place $N$ circles with radius $r$ ($R>r$) on the larger circles circumference equidistantly, so every $2\pi /N$ in the angular sense. What is then the relationship between $R$ and $r$ such that all neighboring circles exactly touch?

I've been trying to write down some equations with arc lengths and such for $N=4$, but I can't seem to get anything sensible out of it.

I've been trying to write down some equations with arc lengths and such for $N=4$, but I can't seem to get anything sensible out of it.

asked 2022-04-06

It is not possible for a part of any of three conic sections to be an arc of a circle.

It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever (algebra). But, to me, this seems to be quite a difficult task. How can this be done?

It is asked to prove this theorem without using any notation or any modern form of symbolism whatsoever (algebra). But, to me, this seems to be quite a difficult task. How can this be done?