Draw a Circle Tangent to a Line and a Circle
A Circle Tangent to a Given Line and a Given Circle
by Fhonda Danley & September Matteson
Trouble: Nosotros are given an an arbitrary line and an arbitrary circle, construct a circle tangent to the given line and the given circumvolve. What is the locus of the centers of all such circles?
First nosotros must find 1 such circle. Let'southward use GSP to discover this circumvolve.
Construction: Construct an arbitrary line and an arbitrary circumvolve. The line is night blue and labled IC. The circle is black and labled D. Nosotros want to find a circumvolve that is tangent to line IC and circumvolve with center D. Commencement find an arbitrary betoken on line IC. Allow's phone call it indicate I. Make a perpendicular line through indicate I and perpendicular to IC. Now find radius of black circle, called FD. Re-create radius onto point I and make a circle with center I. Circle with eye I is congruent to circle with heart D. Lable the intersection on the circle and the perpendicular line, point J and indicate K. Construct segment KD. Perpendiculary bisect KD with a dotted line. Lable the intersection of the perpendicular bisector and line KJ, point Q. Construct line QD. Now form circle with centre Q and radius QI.
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Click here to run across an animatied GSP sketch Definition of Parabola: A parabola is the prepare of all points (x, y) that are equidistant from a fixed line (directrix) and a stock-still point (focus) not on the line.
Using GSP, we can breathing the pink circle and trace the center. We can come across that the locus of the middle of each circle forms a parabola. In the above picture, the parabola is light blue. The directrix is the night bluish line KZ. The focus of the parabola is point D. The altitude from any betoken on the parabola to the directrix is the same equally the altitude from any point on the parabola to the focus. In this particular picture, Q is the bespeak of the parabola, K is an arbitrary point on the dirextrix and D is the focus.
Proof : I desire to prove that the locus of the centers of each circle forms a parabola. The definition of a parabola states that the distance from the directrix to any point on the parabola is the same as the distance from the same indicate on the parabola to the focus. Allow D be the focus and KZ be the directrix. I constructed triangle KQD to be isosceles where
KQ = QD. We tin also see that triangle QNK is coinciding to triangle QND by side angle side. (I perpendicularly bisected KD. So KN is coinciding to DN and bending QNK is coinciding to QND and QN is coinciding to itself). So triangle QNK is congruent to triangle QND. Therefore QK is coinciding to QD. Thus the distance from the directrix to the center of the circumvolve is the aforementioned as the altitude from the center of the circle to the focus. The locus of the centers of the circles form a parabola.
What happens if the tangent surrounds the given circle?
Let's look at another motion picture. 
Construction : Construct an capricious line and an arbitrary circle. The line is black and labeled AF the circumvolve is blackness with center D. Allow AF be an arbitrary point on the line. Drop a dotted blackness perpendicular through AF and perpendicular to the line. Find raius of circumvolve with center D and call it DP. Now form circle with center AF and radius of length DP. Characterization the intersections of the perpendicular line and the circle Af and AJ. Course a line from AJ to D. Perpendicularly bisect this line with a dotted red line. Label the intersection of the red dotted line and the blackness dotted perpendicular AL. Now form lines from AL to D and from AL to AJ. Nosotros accept constructed an isosceles triangle where sides ALD and ALAJ are congruent. Now class a circe with center AL and radius ALAK.
Click here to run across an blithe GSP sketch. Definition of a Parabola: A parabola is the set of all points (x,y) that are equidistant from a fixed line (directrix) and a fixed point (focus) not on the line.
theorize: The foci of the dark-green circles tangent to both the line and the circle forms a parabola. The parabola is blueish. The focus is point D and the directrix is the solid red line.
proof: I want to prove that the locus of the centers of each circle froms a parabola. The difinition of parabola states that the distance from the directrix to any point on the parabola is the same as the distance from the aforementioned betoken on the parabola to the focus. Let D be the focus and the red line passing through the point AJ exist the directrix. Let AL exist an capricious center of a circle tangent to both a line and a circumvolve. I want to prove that AL AJ is the same length every bit AL D. I constructed triangle D AJ AL to be an isosceles triangle. We can show that these two segments are coinciding another way though. I perpendicularly bisected segment AJ d, then AJ AKis congruent to D AK and angle ALAKAJ is congruent to ALAKD. Since ALAK is coinciding to itself, by SAS, triangles ALAKAJ and ALAKD are congruent. Then ALAJ and ALD are coinciding and have the same length. Since they have the same length, we can say that the foci of the circles are the same length from the directrix equally they are from the fucus. Therefor the locus of the centers of the circles course a parabola.
Source: http://jwilson.coe.uga.edu/EMT669/Student.Folders/Matteson.September/tan/tangent.html
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