Exploring Analyic Geometry with Mathematica®

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Tangent Circles

Tangent Object, Center Point
Tangent Object, Center on Object, Radius
Two Tangent Objects, Center on Object
Two Tangent Objects, Radius
Three Tangent Objects
Explorations

In our study of circles we noted that the equation of a circle

"tcir_1.gif"

has three parameters, h, k and r, that may be varied independently. A circle, therefore, is said to have three degrees of freedom (DOF). These degrees of freedom allow us to construct a circle so that it meets three conditions. Some common conditions and the corresponding equations that establish the condition are shown in Table [tcir:tbl01].

"tcir_2.gif"

Circle tangency equations.

By specifying a combination of conditions so that the degrees of freedom add up to three, we can then solve three simultaneous equations in three unknowns (h, k and r) and determine the (possibly empty) set of circles that satisfy the conditions. For economy of expression in the following sections, we will use the convention that a point which is on a circle (i.e. satisfies the equation of the circle) will be said to be tangent to the circle.

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Tangent Object, Center Point [Top]

To construct a circle tangent to an object (a point, line or circle) with a given center point, we select equations as follows from Table [tcir:tbl01]. To establish the condition that a circle be tangent to a point, line or circle, we select the appropriate equation from cases [5], [6] or [7]; to establish the condition that a circle have a given center point we select the two equations from case [1]. Solving these three equations in three unknowns produces the values for the parameters h, k and r of the circles which satisfy the stated conditions.

Example: Construct the circle(s) tangent to the circle "tcir_3.gif" with center point (-1,0) and plot.

Solution: The function TangentCircles2D[{circle},point] returns a list of circles tangent to a circle with a given center point.

cirs=TangentCircles2D[{c1=Circle2D[{0,0},2]},
                      p1=Point2D[{-1,0}]]

"tcir_4.gif"

Sketch2D[{p1,c1,cirs}]

"tcir_5.gif"

Descarta2D Hint: TangentCircles2D[{pt | ln | cir},point] is the general function that returns a list of circles tangent to an object (a point, line or circle) with a given center point. The vertical bar syntax separating the point, line and circle arguments indicates that a point, line or circle may be specified.

Tangent Object, Center on Object, Radius [Top]

To construct a circle tangent to an object (a point, line or circle) with center point on an object (a line or circle), with a given radius, we select equations as follows from Table [tcir:tbl01]. To establish the condition that a circle be tangent to a point, line or circle, we select the appropriate equation from cases [5], [6] or [7]; to establish the condition that the center of the circle be on a given line or circle we select the appropriate equation from cases [3] or [4]; to establish the condition that the circle have a given radius we select the equation from case [1]. Solving these three equations in three unknowns produces the values for the parameters h, k and r of the circles which satisfy the stated conditions.

Example: Construct the circle(s) tangent to the circle "tcir_6.gif", center on the line x-2y+1=0 and with radius 1 and plot.

Solution: The function TangentCircles2D[{circle},line,r] returns a list of circles tangent to a given circle, with center on a line, with a given radius.

cirs=TangentCircles2D[{c1=Circle2D[{0,0},2]},
                       l1=Line2D[1,-2,1], 1]

"tcir_7.gif"

Sketch2D[{l1,c1,cirs}]

"tcir_8.gif"

Descarta2D Hint: TangentCircles2D[{pt | ln | cir},ln | cir,r] is the generalized function that returns a list of circles tangent to an object (a point, line or circle) with center on a line or circle, with a given radius, r.

Two Tangent Objects, Center on Object [Top]

To construct a circle tangent to two objects (points, lines or circles) with center point on an object (a line or circle), we select equations as follows from Table [tcir:tbl01]. To establish the condition that a circle be tangent to a point, line or circle, we select the appropriate equation from cases [5], [6] or [7]-this produces two equations (one for each tangent object); to establish the condition that the center be on a line or circle we select the appropriate equation from cases [3] or [4]. Solving these three equations in three unknowns produces the values for the parameters h, k and r of the circles which satisfy the stated conditions.

Example: Construct the circle(s) tangent to the two circles

"tcir_9.gif"

with center point on the line x-2y+1=0 and plot.

Solution: The function TangentCircles2D[{cir,cir},line] returns a list of circles tangent to two circles, with center point on a given line.

cirs=TangentCircles2D[{c1=Circle2D[{-2,0},1],
                       c2=Circle2D[{2,0},1]},
                       l1=Line2D[1,-2,1]]

"tcir_10.gif"

Sketch2D[{l1,c1,c2,cirs}]

"tcir_11.gif"

Descarta2D Hint: TangentCircles2D[{pt | ln | cir,pt | ln | cir},ln | cir] is the general function that returns a list of circles tangent to two objects (points, lines or circles) with center point on a line or circle.

Two Tangent Objects, Radius [Top]

To construct a circle tangent to two objects (points, lines or circles) with a given radius, we select equations as follows from Table [tcir:tbl01]. To establish the condition that a circle be tangent to a point, line or circle, we select the appropriate equation from cases [5], [6] or [7]-this produces two equations (one for each tangent object); to establish the condition that the circle have a given radius we select the equation from case [1]. Solving these three equations in three unknowns produces the values for the parameters h, k and r of the circles which satisfy the stated conditions.

Example: Construct the circle(s) tangent to the two circles

"tcir_12.gif"

with a radius of 1 and plot.

Solution: The function TangentCircles2D[{circle,circle},r] returns a list of circles tangent to two circles, with a given radius.

cirs=TangentCircles2D[{c1=Circle2D[{-2,0},3],
                       c2=Circle2D[{2,0},3]},1]

"tcir_13.gif"

Sketch2D[{c1,c2,cirs}]

"tcir_14.gif"

Descarta2D Hint: TangentCircles2D[{pt | ln | cir,pt | ln | cir},r] is the general function that returns a list of circles tangent to two objects (points, lines or circles) with a given radius, r.

Three Tangent Objects [Top]

To construct a circle tangent to three objects (points, lines or circles), we select equations as follows from Table [tcir:tbl01]. To establish the condition that a circle be tangent to a point, line or circle, we select the appropriate equation from cases [5], [6] or [7]-this produces three equations (one for each tangent object). Solving these three equations in three unknowns produces the values for the parameters h, k and r of the circles which satisfy the stated conditions.

Example: Construct and plot the circle(s) tangent to the three lines x-y+1=0, x+y-1=0 and y+1=0.

Solution: Use the function TangentCircles2D[{ln,ln,ln}] that returns a list of circles tangent to the three lines.

cirs=TangentCircles2D[{
        l1=Line2D[1,-1,1],
        l2=Line2D[1,1,-1],
        l3=Line2D[0,1,1]}] //Simplify

"tcir_15.gif"

Sketch2D[{l1,l2,l3,cirs},PlotRange->{{-5,5},{-5,5}}]

"tcir_16.gif"

Descarta2D Hint: The function TangentCircles2D[{obj"tcir_17.gif",obj"tcir_18.gif",obj"tcir_19.gif"}] is the general function that returns a list of circles tangent to three objects (points, lines or circles).

Explorations [Top]

Archimedes' Circles. [archimed.html]

"tcir_20.gif"

Draw the vertical tangent line at the intersection point of the two smaller tangent circles, "tcir_21.gif" and "tcir_22.gif", in an arbelos (shoemaker's knife, see figure). Prove that the two circles C' and C'' tangent to this line, the large semicircle, "tcir_23.gif" and "tcir_24.gif" and "tcir_25.gif" are congruent (have equal radii). These circles are known as Archimedes' Circles.

Circle Tangent to Circle, Given Center [tancir1.html]

Show that the radii of the two circles centered at "tcir_26.gif" and tangent to the circle

"tcir_27.gif"

are given by

"tcir_28.gif"

where "tcir_29.gif". This formula is a special case of the Descarta2D function TangentCircles2D[{pt | ln | cir},point].

Circle Tangent to Circle, Center on Circle, Radius. [tancir2.html]

Show that the centers (h,k) of the two circles passing through the point "tcir_30.gif" with center on the circle "tcir_31.gif" and radius r=1 are given by

"tcir_32.gif"

where "tcir_33.gif". This is a special case of the Descarta2D function

TangentCircles2D[{pt | ln | cir},ln | cir,r]

that constructs a list of circles.

Circle Tangent to Two Lines, Radius. [tancir3.html]

Show that the centers (h,k) of the four circles tangent to the perpendicular lines

"tcir_34.gif"

with radius r=1 are given by

(h,k) = "tcir_35.gif"
. = "tcir_36.gif"
. = "tcir_37.gif"
. = "tcir_38.gif"

Assume that the two lines are normalized, "tcir_39.gif". This construction is a special case of the Descarta2D function TangentCircles2D[{obj"tcir_40.gif",obj"tcir_41.gif"},r] when the two objects are lines.

Circle Through Two Points, Center on Circle. [tancir4.html]

Show that the radii of the two circles passing through the points (0,a) and (0,-a) with centers on the circle "tcir_42.gif" are both given by

"tcir_43.gif"

This is a special case of TangentCircles2D[{obj"tcir_44.gif",obj"tcir_45.gif"},ln | cir] where the two objects are points.

Circle Tangent to Three Lines. [tancir5.html]

Show that the radii of the four circles tangent to the lines

x=0,  y=0   and   Ax+By+C=0,

are given by

"tcir_46.gif"

taking all four combinations of signs and assuming the lines are normalized. This is a special case of the function TangentCircles2D[{obj"tcir_47.gif",obj"tcir_48.gif",obj"tcir_49.gif"}] where all three of the objects are lines.

Circles Tangent to an Isosceles Triangle. [tncirtri.html]

A circle is inscribed in an isosceles triangle with sides a, a and 2b in length. A second, smaller circle is inscribed tangent to the first circle and to the equal sides of the triangle. Show that the radius of the second circle is

"tcir_50.gif"

Assume that a>b.


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