Question

$\overleftrightarrow{cd}$ is the bisector of $\angle c$. construct the incenter of $\triangle abc$.

Explanation:

Step1: Recall incenter definition

The incenter of a triangle is the intersection point of its angle bisectors. We know \( \overrightarrow{CD} \) bisects \( \angle C \), so we need another angle bisector (e.g., of \( \angle A \) or \( \angle B \)).

Step2: Choose an angle to bisect

Let's bisect \( \angle B \) (or \( \angle A \)). Using the construction tools (e.g., compass and straightedge, or the given digital tools like 'Ray' for bisecting), construct the angle bisector of \( \angle B \).

Step3: Find intersection

The intersection of the angle bisector of \( \angle B \) (or \( \angle A \)) and \( \overrightarrow{CD} \) is the incenter. Let's denote the incenter as \( I \).

(Note: In a digital construction, using the 'Ray' tool to bisect another angle (e.g., \( \angle B \)): place the compass at \( B \), draw arcs intersecting \( BA \) and \( BC \), then draw a ray from \( B \) through the intersection of the arcs' bisectors. Then find where this ray meets \( CD \).)

Answer:

The incenter is the intersection point of \( \overrightarrow{CD} \) (angle bisector of \( \angle C \)) and the angle bisector of either \( \angle A \) or \( \angle B \). When constructed, it is the point where these two angle bisectors meet (e.g., denoted as \( I \) in the triangle).