Question of Triangle of GeometryIn triangle ABC,we have A B > A C. If A'...

Leila Jennings

Leila Jennings

Answered

2022-07-16

Question of Triangle of Geometry
In triangle ABC,we have A B > A C. If A' is the mid point of BC, AD is the altitude through A and if the internal and external bisectors of angle A meet at BC at X and X' respectively ,prove that A D = ( c 2 b 2 ) / 2 a
I have tried to use obtuse angle theorem to find height then solve the question but stuck on double variable.

Answer & Explanation

bulgarum87

bulgarum87

Expert

2022-07-17Added 15 answers

Step 1
Since c > b, we obtain γ > β ,, which says that A B C is an acute angle.
Thus, B D = c cos β = c ( a 2 + c 2 b 2 ) 2 a c = a 2 + c 2 b 2 2 a .
Step 2
Also, me see that B D > B A because c > b.
Thus, A D = B D B A = a 2 + c 2 b 2 2 a a 2 = c 2 b 2 2 a .

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