A circle is inscribed in the triangle, which touches the sides AB, AC and BC

A circle is inscribed in the triangle, which touches the sides AB, AC and BC, respectively, at points P, F and M. Find the length of the segments AP, PB, BM, MC, CF and FA, if AB = 8 cm, BC = 6 cm, AC = 12 cm.

Let the length of the segment AF = X cm, then the length of the segment CF = (12 – X) cm.

By property of tangents:

AP = AF = X cm, then BP = (8 – X) cm.

CM = CF = (12 – X) cm.

BM = BC – CM = 6 – (12 – X) = X – 6.

Since, by the property of tangents, ВМ = ВР, then:

X – 6 = 8 – X.

2 * X = 14.

X = 14/2 = 7 cm.

AP = AF = 7 cm, then CF = CM = 12 – 7 = 5 cm.

BM = BP = 7 – 6 = 1 cm.

Answer: AP = AF = 7 cm, CF = CM = 5 cm, BP = BM = 1 cm.