Problem 1: Construct an Interior Tangent to Two Circles of the Same Diameter

Draw a horizontal line. Mark two points, O and O', on this line such that the distance between them is $120 \text{ mm}$. These will be the centers of the two circles.

• With O as the center, draw a circle with a radius of $15 \text{ mm}$.
• With O' as the center, draw another circle with a radius of $15 \text{ mm}$.

Find the midpoint M of the line segment OO'. This can be done by drawing arcs of equal radius (greater than half of OO') from O and O', and connecting their intersection points. The line connecting these points will bisect OO' at M.

With M as the center, draw a semicircle with radius MO (or MO'). This semicircle will pass through O and O'.

• From O, draw a line perpendicular to OO' upwards. This line will intersect the semicircle at a point, let's call it X.
• From O', draw a line perpendicular to OO' downwards. This line will intersect the semicircle at a point, let's call it Y.

Draw a straight line connecting point X and point Y. This line XY is the common interior tangent to the two circles.

: Construct an External Tangent Between Two Circles of Different Radii

Given:

• Radius of the first circle $R_1 = 25 \text{ mm}$
• Radius of the second circle $R_2 = 15 \text{ mm}$
• Distance between their centers = $100 \text{ mm}$

Construction Steps:

Draw a horizontal line. Mark two points, O and O', on this line such that the distance between them is $100 \text{ mm}$. O will be the center of the larger circle, and O' for the smaller one.

• With O as the center, draw a circle with radius $R_1 = 25 \text{ mm}$.
• With O' as the center, draw a circle with radius $R_2 = 15 \text{ mm}$.

Calculate the difference in radii: $R_{diff} = R_1 - R_2 = 25 \text{ mm} - 15 \text{ mm} = 10 \text{ mm}$. With O as the center, draw a third (construction) circle with radius $R_{diff} = 10 \text{ mm}$.

Find the midpoint M of the line segment OO'.

With M as the center, draw a semicircle with radius MO (or MO'). This semicircle will pass through O and O'.

From O', draw a line perpendicular to OO' that intersects the semicircle at a point, let's call it P.

Draw a straight line from O to P. This line OP is tangent to the difference circle (the $10 \text{ mm}$ radius circle).

Extend the line OP until it intersects the larger circle (radius $R_1 = 25 \text{ mm}$) at a point. Let's call this point T1. This is the tangent point on the larger circle.

From O', draw a line parallel to the line segment OT1. This parallel line will intersect the smaller circle (radius $R_2 = 15 \text{ mm}$) at a point. Let's call this point T2. This is the tangent point on the smaller circle.

Draw a straight line connecting point T1 and point T2. This line T1T2 is the common external tangent to the two circles.