Suppose that c is a positive integer. Define f(c) to be the number of pairs (a, b) of positive integers with c < a < b for which two circles of radius a, two circles of radius b, and one circle of radius c can be drawn so that • each circle of radius a is tangent to both circles of radius b and to the circle of radius c, and • each circle of radius b is tangent to both circles of radius a and to the circle of radius c, as shown. Determine all positive integers c for which f(c) is even.
Angles in Circles
Angles within a circle are feasible to create with the help of different properties of the circle such as radii, tangents, and chords. The radius is the distance from the center of the circle to the circumference of the circle. A tangent is a line made perpendicular to the radius through its endpoint placed on the circle as well as the line drawn at right angles to a tangent across the point of contact when the circle passes through the center of the circle. The chord is a line segment with its endpoints on the circle. A secant line or secant is the infinite extension of the chord.
Arcs in Circles
A circular arc is the arc of a circle formed by two distinct points. It is a section or segment of the circumference of a circle. A straight line passing through the center connecting the two distinct ends of the arc is termed a semi-circular arc.
Suppose that c is a positive integer. Define
f(c) to be the number of pairs (a, b) of
positive integers with c < a < b for which
two circles of radius a, two circles of radius b,
and one circle of radius c can be drawn so
that
• each circle of radius a is tangent to both
circles of radius b and to the circle of
radius c, and
• each circle of radius b is tangent to both
circles of radius a and to the circle of
radius c,
as shown. Determine all positive integers c
for which f(c) is even.
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