SOLUTION: A and B are on the same side of a line L. AD and BE are perpendiculars drawn to the line L. If C is the midpoint of AB then prove that CD = CE.

  

  

The plot for this problem is shown in the Figure on the right.

We need to prove that two red colored segments are congruent.

Let us draw the perpendicular CF (green line) from the point C to the line DE.
The perpendicular CF is parallel to both lines AD and BE.

Thus we have three parallel lines AD, CF and BE, and two transverses DE and AB.

Now apply this property:

    If three parallel lines cut off two congruent segments in one transverse line, 
    then they cut off two congruent segments in any other transverse line.

(See the lesson 
Three parallel lines cutting off congruent segments in a transverse line 
in this site).
 
  

      


                     Figure
 
 

Since the segments AC and CB are congruent, the segments DF and FE are congruent too, according to this property.

Now we have two right-angled triangles DCF and ECF.
They have congruent legs DF and EF and the common leg CF.
Hence, these triangles are congruent according to SAS-test for triangles congruency.

It implies that CD = CE as the corresponding sides of congruent triangles.

The proof is completed.