Question

14 use a compass and straightedge to construct the incenter of δabc. 15 construct the angle bisector of each given angle. (a) (b)

Explanation:

Step1: Recall Incenter Definition

The incenter of a triangle is the intersection of its angle bisectors. So, we need to construct two angle bisectors (of two angles of \( \triangle ABC \)) and find their intersection.

Step2: Construct Angle Bisector of \( \angle A \)

• Place the compass tip at \( A \). Draw an arc that intersects \( AB \) and \( AC \) at points \( D \) and \( E \), respectively.
• Without changing the compass width, place the tip at \( D \) and draw an arc inside the triangle. Then place the tip at \( E \) and draw another arc intersecting the previous arc at \( F \).
• Use the straightedge to draw \( \overrightarrow{AF} \), the angle bisector of \( \angle A \).

Step3: Construct Angle Bisector of \( \angle B \)

• Place the compass tip at \( B \). Draw an arc that intersects \( BA \) and \( BC \) at points \( G \) and \( H \), respectively.
• Without changing the compass width, place the tip at \( G \) and draw an arc inside the triangle. Then place the tip at \( H \) and draw another arc intersecting the previous arc at \( I \).
• Use the straightedge to draw \( \overrightarrow{BI} \), the angle bisector of \( \angle B \).

Step4: Find Intersection

The intersection point of \( \overrightarrow{AF} \) and \( \overrightarrow{BI} \) is the incenter of \( \triangle ABC \). (We could also construct the angle bisector of \( \angle C \) for verification, but two are sufficient to find the intersection.)

Answer:

The incenter is the intersection of the angle bisectors of \( \triangle ABC \), constructed by bisecting two angles (e.g., \( \angle A \) and \( \angle B \)) as described, and finding their intersection point. (For the angle bisector construction in 15, follow similar steps: for an angle with vertex \( X \) (or \( A \)), bisect by drawing arcs from the sides, intersecting arcs, and drawing the bisector.)