Question

a circle with centre p inscrubed in triangle ABC side AB side BC side AC touches circle at point L M N respectively radius of circle is r prove that area of triangle ABC is 1/2 × ab + bc + ac ×r​

Answer 1

Answer:

a circle with Centre P is inscribed in a triangle ABC side a b side BC and side AC touch the circle at points L,m and n respectively the radius of the circle is r

prove that area of triangle ABC =1/2(AB+BC+AC)×r

Let say center point = O

if we draw line from points A , B & C at point O

we can Divide ΔABC into three triangle

ΔAOB , ΔBOC & ΔCOA

Area of ΔAOB = (1/2) * AB * OL ( Base * Perpendicular)

OL = Radius = r

Area of ΔAOB = (1/2) * AB * r

SImilarly

Area of ΔBOC = (1/2) * BC * r

Area of ΔBOC = (1/2) * BC * r

Area of ΔCOA = (1/2) * AC * r

Area of ΔABC = Area of ΔAOB + Area of ΔBOC + Area of ΔCOA

=> Area of ΔABC = (1/2) * AB * r + (1/2) * BC * r + (1/2) * AC * r

=> Area of ΔABC = (1/2) * (AB + BC + AC) * r

QED