the bisector of angle b and angle ç of triangle ABC meet at opposite side in point d and e respectively if ed is parallel to BC prove that triangle ABC is an isosceles triangle
Answers
Given : bisector of angle b and angle ç of triangle ABC meet at opposite side in point d and e respectively
ED || BC
To Find : prove that triangle ABC is an isosceles triangle
Solution:
Let say ∠B = 2x and ∠ C = 2y
BD bisector of ∠B
Hence ∠CBD = x and ∠EBD = x ( as E lies on AB)
CE is bisector of ∠C
=> ∠BCE = y and ∠DCE = y ( as E lies on AB)
ED || BC
=> ∠EDB = ∠DBC = x
Hence ∠EBD = ∠EDB
=> EB = ED ( sides opposite to Equal angles)
Similarly ED = DC
=> EB = DC
as ED || BC
=> AE/EB = AD/DC ( BPT)
=> EB = DC
=> AE = AD
AE + EB = AD + DC
=> AB = AC
AB = AC
Hence triangle ABC is isosceles triangle
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