Properties of TriangleHard

Question

Internal bisector of ∠A of triangle ABC meets side BC at D. A line drawn through D perpendicular to AD intersects the side AC at E and the side AB at F. If a, b, c represent sides of ᐃABC then

Options

A.AE is HM of b and c
B.AD =
C.EF =
D.the triangle AEF is isosceles

Solution

       
We have ᐃABC = ᐃABD + ᐃACD
⇒   bc sin A = cAD sin b × AD sin
⇒   AD =
Again AE = AD sec
= ⇒ AE is HM of b and c.
EF = ED + DF = 2DE = 2 × AD tan
=
As AD ⏊ EF and DE = DF and AD is bisector ⇒ AEF is isosceles.
Hence A, B, C and D are correct answers.

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