So many triangles?
Let be the total number of isosceles and equilateral triangles with integer sides such that no side exceeds 2016. What is ?
The answer is 1512.
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1 solution
If the sides are a,a,b then the triangle is formed only when 2a>b.So for any Natural number a , b can change from 1 to 2a-1 .
case 1 : a<= 1008 . No. of isosceles triangles = 1+3+5+............+2015 = 1008x1008 . case 2: 1009<=a<=2016 .
No. of isosceles triangles = 1008x2016 . Therefore, the total number of isosceles triangles = 3x1008x1008 . .
Remark:The above count of isosceles triangles includes equilateral triangles also.