In the adjoining figure, two points A and B lie on the same side of a line XY. If `AD bot XY, BE bot XY` and C is the midpoint of AB, prove that CD = CE.

Explanation

Draw `CM bot XY.`
Now, `AD bot XY, CM bot XY and BE bot XY rArr `AD||CM||BE.
Thus, AD, CM, BE are three parallel lines, cut by the transversal ACB at A, C and B respectively such that AC = CB.
These lines AD, CM, BE are also cut by the transversal XY at D, M and E respectively.
`therefore DM = ME " " ` (by intercept theorem).
Now, in `triangle `CDM and CEM, we have
`DM = ME " (proved), "CM=CM " (common)," `
`angle CMD = angle CME = 90^(@).`
`therefore triangle CDM ~= triangle CEM.`
Hence, `CD = CE " " `(c.p.c.t.).