We have been told that two circles having only one point in common are tangent circles.
To draw three tangent circles that are centered at each vertex of any triangle, we will begin by finding the incenter of the arbitrary triangle △ ABC.
Incenter of △ ABC
Recall that the point of intersection of the angle bisectors of a triangle is called the incenter of the triangle. Therefore, to find the incenter of △ ABC, we will first find the angle bisector of ∠ A. To do so, we put the compass point on vertex A, draw an arc that intersects sides of ∠ A, and label the intersection points.
Next, we will put the compass point on point X and draw an arc. With the same compass setting, draw another arc by using point Y such that the arcs intersect. Then, we will label the point of intersection.
Next, we will draw AZ which is the bisector of ∠ A.
Proceeding in the same way, we will draw the bisectors of ∠ B and ∠ C. Then, we will label the incenter of △ ABC.
Tangent Circles
Since we found the incenter of △ ABC, we can draw the inscribed circle of the triangle that is tangent to the sides. Then, we will label the points of tangency.
Now that we have the inscribed circle of triangle, let's recall Theorem 12-3.
If two tangent segments to a circle share a common endpoint outside the circle, then the two segments are congruent.
By this theorem, we can conclude that AF=AE, BF=BD, and CD=CE.
Now, let's consider AF and AE as radii of the circle which is centered at vertex A. With this, we can draw the first circle.
Finally, proceeding in the same way, we can draw the other two tangent circles.
As we can see, the circles are tangent to each other at D, E, and F. Note that this method of construction is valid for any triangle.
