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Incentroid (Posted on 2019-02-24) Difficulty: 3 of 5
The segment formed by joining the incenter and centroid of a certain right triangle is parallel to one of its legs and has a length of 1 unit.

Find the perimeter of this triangle.

No Solution Yet Submitted by Danish Ahmed Khan    
Rating: 3.0000 (1 votes)

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Solution | Comment 1 of 2
The centroid has the property that its distance to the vertex of the triangle is twice the distance to the opposite edge. 

Knowing this this is an easy problem because: 

Be b the base and h the height of the triangle, r the radio of the inscribed circle. You have:

(h/3)-(b/3)=1
b/3=r
h/3=r+1

In addition if we call (ipsilon)=E half the angle of the triangle which is closer to the incenter and is not the right angle, then:

tan(E)=r/(2/3*b)=1/2 and
tan(2E)=h/b=(r+1)/r

(but we know tan(E)=1/2 then tan(2E)=4/3, so=

(r+1)/r=4/3  => r=3

Then b=9 and h=12 

It is right triangle with edges 9,12,15 and the perimeter is
p=36

It could be possible (and perhaps more interesting) to do this problem with the segment parallel to the hypotenuse, bur when you solve the equations for it, no real value fits. 

Edited on March 1, 2019, 3:38 pm
  Posted by armando on 2019-03-01 09:21:25

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