Question

given: see the diagram. prove: dc = de
statement reason

1. $overleftrightarrow{dc}perpoverline{ac}$ given
2. ag = gc given
3. $overleftrightarrow{dc}$ is the perpendicular bisector of $overline{ac}$ deduced from steps 1 and 2
4. da = dc
5. $overleftrightarrow{dh}perpoverline{ab}$ given
6. ah = hb given
7. $overleftrightarrow{dh}$ is the perpendicular bisector of $overline{ab}$ definition of perpendicular bisector
8. da = db deduced from steps 6 and 7
9. dc = de transitive property of equality

Explanation:

Step1: Define perpendicular bisector property

A point on the perpendicular bisector of a line - segment is equidistant from the endpoints of the line - segment. Since $\overrightarrow{DG}\perp\overrightarrow{AC}$ and $AG = GC$, $D$ is on the perpendicular bisector of $\overrightarrow{AC}$, so $DA=DC$.

Step2: Apply the same property

Since $\overrightarrow{DH}\perp\overrightarrow{AB}$ and $AH = HB$, $D$ is on the perpendicular bisector of $\overrightarrow{AB}$, so $DA = DB$.

Step3: Use the transitive property

We know that $DA=DC$ and $DA = DB$. By the transitive property of equality (if $a = b$ and $a = c$, then $b = c$), we can conclude that $DC=DB$.

Answer:

The proof that $DC = DB$ is completed by using the property of the perpendicular bisector (a point on the perpendicular bisector of a line - segment is equidistant from the endpoints of the line - segment) and the transitive property of equality.