AD and AE are respectively the altitude and angle bisector of ΔABC (D and E are on side BC). If DB - DC = 1029 and EB - EC = 189 then what is the value of AB - AC?
This is a not-so-easy problem with an easy solution
From the initial data it is easy to determine the distance between E and D. ED=420.
Now we have two different sources of data on triangle ABD:
- A-BD-C is ABC divided by the altitude AD
- A-BE-C is ABC divided by the bisector AE
We can work on each of them trying to interlace the results.
To work on them we choose to rely on two angles alfa (a) and gamma (g):
- alfa (a) is the angle on vertex A opposite to BE
- gamma (g) is the angle on vertex A opposite to ED
With this angles all the segments of BC can be reached, because BE is opposite to a, BD is opposite to (a+g), ED is opposite to g, and DC is opposite to (a-g).
Now we try to express DB, DC, EB, EC in terms of a e g.
DB= AD*tan(a+g) = 420*(tan(a+g)/tan(g))
DC= AD*tan(a-g)= 420*(tan(a-g)/tan(g))
So DB-DC=1029=420*((tan(a+g)-tan(a-g))/tan(g)
In a similar way it is posible to operate with bisector (here you need a little more work) and you get:
EB-EC=189=420*tan(a)*((tan(a+g)-tan(a-g))
Solving the system you get:
tan(a)=3/7
tan(g)=3/7
This implies that DC=0 and that the triangle ABC is a straight triangle.
From here you get easily:
AB=1421 AC=980 BC=BD=1029
And AB-AC=441= 21^2
This surprising "outcome" can explain the title?
Edited on February 26, 2019, 11:19 am
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Posted by armando
on 2019-02-26 03:15:54 |
Solution
