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 { a c } \\) 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 a b c \\). make a conjecture about the excircle and the sides of \\( \triangle a b c \\).

Explanation:

• For the first conjecture: The intersection point \(P\) of the angle bisectors (the bisector of \(\angle A\), and the bisectors of the exterior angles \(\angle BCE\) and \(\angle CBD\)) is the ex - center of \(\triangle ABC\). An ex - center is equidistant from the sides of the triangle (one side and the extensions of the other two sides).
• For the second conjecture: The excircle (with center \(P\) and radius \(PM\)) is tangent to side \(AC\) (since \(PM\perp AC\) and \(PM\) is the radius) and is also tangent to the extensions of sides \(AB\) and \(BC\). This is based on the property that the distance from the ex - center to each of the relevant sides (one side of the triangle and the extensions of the other two) is equal (the radius of the excircle).

Answer:

• First conjecture: The intersection of the angle bisectors is the ex - center of \(\triangle ABC\).
• Second conjecture: The excircle is tangent to side \(AC\) and the extensions of sides \(AB\) and \(BC\).