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# AD and BC are equal perpendiculars to a line segment AB (see figure). Show that CD bisects AB.

In $$\triangle{AOD}$$ and $$\triangle{BOC}$$, we have,

Now, $$\angle{AOD}$$ = $$\angle{BOC}$$ ...(vertically opposite angles)

Also, $$\angle{DAO}$$ = $$\angle{CBO}$$ = $$90^\circ$$

and BD = BC ...(shown in the figure)

Therefore, $$\triangle{AOD}$$ $$\displaystyle \cong$$ $$\triangle{BOC}$$ ...(SAS congruency test)

Hence, OA = OB ...(By CPCT)

Thus, we can say that, O is the mid-point of AB.

So, CD bisects AB.
Hence, proved