ABC is a triangle, the incircle touches the sides BC, CA and AB at D, E and F respectively. BD, CE and AF are consecutive natural numbers. I is the incentre of the triangles. The radius of the incircle is 4 units.

ABC is a triangle, the incircle touches the sides BC, CA and AB at D, E and F respectively. BD, CE and AF are consecutive natural numbers. I is the incentre of the triangles. The radius of the incircle is 4 units. The sides of the triangle are  13,14 and 15 unit ,prove

Solution :
Step1:Follow this figure

Step2:You can easily find that BC=2X+1,AC=2X+3 and AB=2X+2.....1
Step3:Find the area of ABC triangle by Heron's formula =[(3x+3)(x)(x+1)(x+2)]^(1/2).....2
Step4:Add area of three triangles namely BDC,CDA and ABD by using their base and perpendicular =12(x+1) square unit......3
Step5:(3x+3)(x)(x+1)(x+2)=144(X+1)^2........4
Step6:Solving 4 ,would give x=6,the sides of triangles are proved to be 13,14 and 15(Answ)