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n triangle ABC, AB=13, AC=14 and  BC=15. Let I be the incenter. The incircle of triangle ABC  touches sides BC, AB, and AC at D, E, and F, respectively. Find the area of quadrilateral AEIF.

 

 Mar 11, 2020
 #1
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 Mar 11, 2020
edited by Melody  Mar 11, 2020
 #2
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An triangle ABC, AB=13, AC=14 and  BC=15. Let I be the incenter. The incircle of triangle ABC  touches sides BC, AB, and AC at D, E, and F, respectively. Find the area of quadrilateral AEIF.

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 Mar 11, 2020
 #3
avatar+531 
+2

In triangle ABC, AB=13, AC=14 and  BC=15. Let I be the incenter. The incircle of triangle ABC  touches sides BC, AB, and AC at D, E, and F, respectively. Find the area of quadrilateral AEIF.

 

AB = 13        AC = 14       BC = 15

I     c     I       I      b     I      I     a     I

 

Semiperimeter    s = ( a + b + c )/2 = 21

 

Triangle (ABC) area is:    A = sqrt [ s (s-a)(s-b)(s-c)]  = 84 u²

 

Radius   r = A / s   = 4

 

Angle   BAC =>           cos(∠A) = ( b² + c² - a² ) / 2bc   = 67.38°

 

AE = AF  = r / tan (∠A/2)  = 6

 

Area of quadrilateral  AEIF  = AF * r  = 24 u²   indecision

 Mar 12, 2020
edited by Dragan  Mar 12, 2020
edited by Dragan  Mar 12, 2020
edited by Dragan  Mar 16, 2020

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