+83 In parallelogram ABCD point K belongs to diagonal BD so that BK:DK=1:4. If the extension of AK meets BC at point E, what is the ratio of BE:EC?
+129850 Let ABCD be the parallelogram
Angle BKE = Angle AKD (vertical angles)
Since AB is parallel to CD and BD is a transversal cuting these parallels, then
Angle KBE = Angle ADK
Then ΔAKD is similar to ΔEKB
And
BK / DK = BE / AD
But since we have a paralleogram, AD = BC
So
BK / DK = BE / BC
But BK /DK = 1/4
So
BE / BC = 1/4
So
BC / BE = 4/1
[ BE + EC ] / BE = 4
BE / BE + EC / BE = 4
1 + EC / BE = 4
EC / BE = 3
EC / BE = 3/1
So
BE / CE = 1 / 3
