practice problems pending

Q1. Prove that if the two angle bisectors of a triangle are equal, then the triangle is isosceles. (Steiner-Lehmus Theorem)
S1.


So, let's see what are we trying to prove.
BE = CF and angle B = angle C

We know that length of angle bisector in triangle is given by the below formula.
BE = 2ac.cos(B/2)/(a+c)
Similarly,
CF = 2ab.cos(C/2)/(a+b)

make the equal:
c.cos(B/2)/(a+c) = b.cos(C/2)/(a+b)
cos(B/2)/cos(C/2) = ab + bc/ac + bc

We will prove by contradiction.
WLOG, Angle B > Angle C => b > c
also => cos(B/2) < cos(C/2) (since B/2 and C/2 are both < 90).
So LHS < 1
But RHS > 1
why?
Because  ab + bc > ac + bc (since b > c)

And if Angle C < Angle B, again LHS > 1 and RHS < 1.
So only option is that both are equal.
H.P.


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