Question

14 use a compass and straightedge to construct the incenter of δabc.

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 (e.g., of $\angle A$ and $\angle B$) and find their intersection.

Step2: Construct Angle Bisector of $\angle A$

• Place the compass vertex at $A$. Draw an arc that intersects $AB$ and $AC$ at two points, say $D$ (on $AB$) and $E$ (on $AC$).
• Without changing the compass width, place the vertex at $D$ and draw an arc inside the triangle. Then place the vertex at $E$ and draw another arc (with the same width) intersecting the first arc at point $F$.
• Use the straightedge to draw ray $AF$; this is the angle bisector of $\angle A$.

Step3: Construct Angle Bisector of $\angle B$

• Place the compass vertex at $B$. Draw an arc that intersects $BA$ and $BC$ at two points, say $G$ (on $BA$) and $H$ (on $BC$).
• Without changing the compass width, place the vertex at $G$ and draw an arc inside the triangle. Then place the vertex at $H$ and draw another arc (with the same width) intersecting the first arc at point $I$.
• Use the straightedge to draw ray $BI$; this is the angle bisector of $\angle B$.

Step4: Find Intersection

The intersection point of $AF$ and $BI$ is the incenter of $\triangle ABC$. (Note: Constructing a third angle bisector can verify, but two are sufficient for intersection.)

Answer:

The incenter is constructed by finding the intersection of two angle bisectors (e.g., of $\angle A$ and $\angle B$) using compass - straightedge angle - bisecting constructions as outlined. The final incenter is the point where the two constructed angle bisectors meet.