Draw Circles of Different Diameters
Tangent Circles of Two Separate Circles
Brian Swanagan
We will investigate the tangent circles to two circles that share no points within and on their boundaries. I have fatigued two circles of different size below.
At present we cull a point E on our larger circle (or our smaller circle) and create the circle tangent the that point and tangent to the other circumvolve. The first tangent circumvolve I wish to create will non contain either circle so I'd similar to pick a point closer to the space betwixt them. I then draw a circumvolve around this point E identical to the smaller circle. EE' is the same vector as CD. The tangent circle'southward center is lies on a line through E and A so I draw that line to requite an idea of where the heart could lie. We label F which is the intersection between our minor circle copy and the line that lies inside the larger circle.
The radius of our tangent circle plus the radius of the smaller circle is the altitude from the center of the smaller circle to the center of our tangent circle. The altitude from the tangent circle'south center from F is likewise the sum of the radii of the tangent and smaller circle so C, F, and the centre of the tangent circle class an isosceles triangle. CF is the base of this triangle so nosotros connect C and F with a segment and notice its midpoint Thou. Drawing a perpendicular bisector of CF through 1000 gives us a line that the center of the tangent circumvolve lies on so nosotros tin can find the intersection of our ii lines to find the location of the eye of our tangent circumvolve. We may run into a problem when the ii lines are parallel and we volition hash out this case a bit later on. Now, we tin can draw our circle (ruddy) since nosotros have the heart, H, and a point on the circumvolve, E.
As we motility further away from the smaller circle, the tangent circle begins to increment in size and the bending HFC increases in mensurate
Every bit EFC continues to increase in measure out and approaches 90 degrees, the two lines begin to wait more parallel so H moves quite far away from the two circles making the tangent circle enormous. We tin see that it begins to wait similar a line itself relative to our smaller circles in the left film. In fact, the part of the curve of the tangent circle closest to our first two circles begins to approach ane of our tangent lines for our two circles. As E continues to move farther abroad from C, EFC becomes an obtuse angle and the centre for our circle moves to the other side of our starting circles. Now, the circle is tangent to both just containes both circles. This is another set of circles tangent to our original circles a bit different from what we created originally.
As Eastward moves opposite to C, EFC becomes a line or a 180 degree bending and our two lines get perpendicular. H at this betoken has passed through one of our circles or between them. Now, our tangent circle has decreased in size as far as it can while still containing the two circles.
The tangent circle begins to grow again every bit E moves dorsum around and closer to C at present. EFC continues to increase in measure if nosotros look at the larger part of the angle. At some bespeak, EFC reaches 270 degrees and our lines become parallel once again. Now the red bend of our circle approaches another tangent line of our two starting circles.
As E completes its excursion and moves closer to C, the tangent circle decreases until H lies betwixt them both and the circumvolve once again does not comprise the two starting circles.
Next, we motility E near its original position and begin to wait more than closely at the locus of eye points for our tangent circles that we created during this cycle. We connect C and H with a segment beginning and the point I which lies on the intersection betwixt our tangent circle and our smaller circle.
We know that HE = HI and EF = IC so HA - HC = FA which is a abiding value considering FA = R1 - R2 where R1 is the radius of our larger circle and R2 is the radius of our smaller circle. R1 - R2 = 0 if the ii circles are the same size. That ways the loci for the tangent circles that affect just practise not contain the solid green circles should be a hyperbola since the distances from its heart to C and A has a abiding departure.
Tracing F gives us a circle around A and tracing H gives us two hyperbolic looking curves. We may wish to look more closely at the situaiton when the tangent circle encloses our two solid green circles.
How-do-you-do = HE because they are both radii of our tangent circumvolve. Merely, we know CI = Iron are equal to the radius of our smaller circle. That means HC = HF and HF = HA + AF so HC - HA = AF which is once more our constant difference that we found before. And more than interestingly, HA and HC are the distances from the centers of our two circles. This means the loci of the centers of our tangent circles forms a hyperbola with the centers of our 2 circles equally its foci whose difference from the tangent circles is always a constant altitude.
During our final structure, we created tangent circles that either didn't enclose either of the 2 circles or enclosed both. We will now attempt to construct the remaining set of tangent circles that encloses exactly ane of either circumvolve. And so, we construct two circles separate from each other and pic a point E on the larger of the two circles or one of the circles if they are equal. Nosotros then depict a replica of the other circumvolve with Eastward as its heart. EE' and CD are the same vector.
Now, we will intially attempt to create a tangent circle that encloses the larger circumvolve but not the smaller ane so we will move E so that information technology is not likewise shut to C so that it has to enclose the larger circumvolve in society to be tangent to both. The tangent circumvolve is tangent at E so its center must lie on the line through A and E. The heart of our tangent circle to Eastward will once more be equal to the distance to its tangent point on our smaller circle. The radius of our smaller circles plus the radius of the tangent circumvolve gives us to equal distances so we need to mark the intersection of the second smaller circumvolve and the line that is outside the larger circumvolve. Then C, J, and the center of the tangent circumvolve forms an isosceles triangle with CJ as its base so we connect CJ, notice the segments midpoint, and draw the perpendicular bisector to CJ to give us some other line that the center of the tangent circle lies on. When the two lines aren't parallel which only happens in two specific instances that we shall discover later, the intersection of the two lines gives us the middle to our tangent circle that we will call L.
Now, we tin can depict our circle that passes through E and has its heart at L. Equally E moves closer to the smaller circle, angle KJE grows as does the size of our tangent circle.
As the angle measure of KJE approaces xc degrees, the ii lines become more than and more than parallel and and so the curve of our tangent circles approaches that of one of the tangent lines of our 2 circles that passes between them. Every bit E continues to motion closer to C, our tangent circumvolve decreases in side again and now encloses but the smaller circumvolve creating the last ready of tangent circles.
Every bit E continues on its circuit and moves farther away from C, the tangent circle begins to increase in size once again. KJE at present becomes a 270 degree angle making our 2 lines parallel once again and the curve of our tangent circle approaches the fourth and last tangent line of our circles and the 2d that passes between them.
As Due east finishes its motion, the tangent circle now in one case again encloses simply the larger circle completing our set of tangent circles that enclose only ane of the 2 solid green circles.
Now, we would similar to determine the nature of the loci of centers of our tangent circles that enclose exactly one of the solid light-green circles. LJ = CL as nosotros said earlier because they are both the sum of the lengths of the radii of our smaller circle and our tangent circle. LJ = LA + AJ and then CL - LA = AJ and AJ = R1 + R2 where R1 and R2 are the lengths of the radii of our larger and smaller circles, respectively. So, again we accept the divergence of the distances from the centers of our green circles to the tangent circumvolve is a abiding again which should requite a hyperbolic bend.
:Let's make certain our curve is still hyperbolic in nature when the tangent circle only encloses the smaller circumvolve. Again, CL = LJ considering CL plus the radius of our smaller circumvolve and LJ + EJ (the radius of our smaller circle) are both the radius of our tangent circumvolve. LA = AJ + LJ and CL = LJ so LA - CL = AJ our constant divergence once again.
In both cases, the difference of the distances from the centers of our solid green circles to the eye of our tangent circle is always constant so nosotros take a hyperbolic curve if we trace the center, 50, of our tangent circumvolve (traced in black). We should likewise have a circle if nosotros trace our constant lengthed segment AJ (traced in red). Indeed, we see that this is the case.
This means that the loci of the centers of the tangent circles for two split up circles is hypberolic with the foci at the center of our two dissever circles.
Source: http://jwilson.coe.uga.edu/EMAT6680Su06/Swanagan/Assignment7/BSAssignment7.html
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