a cirlce is inscribed in a triangle ABC with P , R , Q as the mid points if side AB , BC , AC respectively . and r is radius of the circle .

prove that

1. AB + CQ = AC +BQ
2. ar of triangle ABC = 1/2 (Perimeter of triangle ABC) * r

given: P, Q and R are the midpoints of the sides AB , BC and AC respectively.

r is the radius of the in circle of triangle ABC.

proof:

(i)

AP = AR........(1) [ the lengths of the tangents drawn from an external point to the circle are equal]

2AP = 2 AR 

AB = AC......(2) [P and R are the mid points of AB and AC]

CQ = BQ .......(3)[Q is the mid point of BC]

adding (2) and (3)

AB +CQ = AC+BQ

which is the result (1)

2.

let the center of the circle be O.

line joining the point of contact to the center is perpendicular to the tangent.

therefore OP⊥AB , OQ⊥BC and OR⊥AC

and OP = OQ = OR = radius of the circle

area (ΔABC) = area(ΔAOB)+area(ΔBOC)+area(ΔCOA)

= 1/2 (perimeter of the triangle ABC)*r

hope this helps you