Question

you can construct an excenter of a circle, in a similar way to constructing the incenter.

step 1: construct the bisector of \\( \angle a \\).

step 2: construct the bisectors of \\( \angle bce \\) and \\( \angle cbd \\).

make a conjecture about the intersection of the angle bisectors.

label the intersection of the angle bisectors \\( p \\).

step 3: construct a line segment perpendicular to \\( \overline{ac} \\) through point \\( p \\). label this point \\( m \\).

step 4: construct a circle with center \\( p \\) that passes through \\( m \\).

the circle is called an excircle of \\( \triangle abc \\). make a conjecture about the excircle and the sides of \\( \triangle abc \\).

Explanation:

Step1: Analyze the intersection of angle bisectors

The intersection of the angle bisectors (in this case, the bisector of \( \angle A\), and the bisectors of the exterior angles \( \angle BCE\) and \( \angle CBD\)) is the ex - center of the triangle.

Step2: Analyze the excircle and the sides of the triangle

Since we constructed a perpendicular from the ex - center \(P\) to side \(AC\) (point \(M\)) and then made a circle with center \(P\) passing through \(M\). By the property of angle bisectors (a point on an angle bisector is equidistant from the sides of the angle), the ex - center is equidistant from the sides of the triangle (one side \(AC\) and the extensions of the other two sides \(AB\) and \(BC\)).

Answer:

1. The intersection of the angle bisectors is the ex - center of \(\triangle ABC\).
2. The excircle is tangent to one side of \(\triangle ABC\) (\(AC\)) and tangent to the extensions of the other two sides (\(AB\) and \(BC\)).