# Parallel segments between 2 circles

#### BorisGrushenko

Hello!

I have an interesting task i am trying to solve.

Given are two circles with centers. A segment a is given too. We have to find all parallel segments to a with same length as a, such that on end of these segments lies on Circle 1 and the other end lies on Circle 2.

Kind of tricky for me. Do you have any suggestions?
Thanks!
Boris

Edit: This is basic geometry, so all we can use is a compass to draw circles and a ruler.

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#### bjhopper

When two circles intersect the common chord produces two circle segments.
Why is this chord perpendicular to the line of centers ?
Construct additional chords parallel to this chord using only ruler and compass.

#### BorisGrushenko

Hey!
Why is this chord perpendicular to the line of centers ?
Well, because the chord represents the third side of an isosceles triangle where the line of centres is the height of that triangle (the equal sides are both radius of circle 1 or circle 2, if you set one of the centers as third point of that triangle). Nice property btw. Didnt see that. But ok, i still dont get, why this property should help me to solve my problem.

Construct additional chords parallel to this chord using only ruler and compass.
Yes, i could do that. But the segment a is not parallel to the line of centres, thus it is not perpendicular to that chord or parallels of that chord.

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#### BorisGrushenko

Hello!
I found a solution i think.

The hint is to work with translations. Define translations with length of segment a and with both directions according to a. Translate the centers in both direction a redraw the circles everytime you do a translation.

If anything works, i will post a graphical solution later.

Thanks!

#### bjhopper

There are two isosceles triangles and the line of centers is the perpendicular bisector of the chord.
Extend the line of centers to complete both diameters.
Strike arcs from each center to find end points of new chords of both circles.
Connect the new chords with ruler.
All chords are parallel.

#### BorisGrushenko

Hello bjhopper,

yes, you are right. Like that you get parallel chords. But how does that fit with the task?

#### bjhopper

There are two chords equal in length to the common chord
To find them think rectangles.

#### BorisGrushenko

There are two chords equal in length to the common chord
To find them think rectangles.
Here they are

still no clue where you want to get me

EDIT: Btw, this is how it looks like after working with translations v_1 and v_2. The asked segments are marked red.

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#### bjhopper

Here they are

still no clue where you want to get me

EDIT: Btw, this is how it looks like after working with translations v_1 and v_2. The asked segments are marked red.

Sorry. I am thinking circle segments not line segments

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#### BorisGrushenko

Oh, ok. Sorry, maybe i used wrong words to describe the task (since i am not native english and especially i am not familiar with the use of technical language).