A circle is inscribed in triangle ABC that touches sides AB, BC and CA at points P, Q and R.

A circle is inscribed in triangle ABC that touches sides AB, BC and CA at points P, Q and R. Find AP if AB = 81 cm, BC = 120 cm, CA = 67 cm.

1) the segments of the tangents are equal. Ap = ar; rc = cq; bq = BP.
Let’s introduce variables
Ap = x; rc = z; bq = y
2) from the condition we compose the system
X + y = 81
Z + x = 67
Y + z = 120

X = 81-y
Z = 67-x

Y + (67- (81-y)) = 120
Y + 67 – 81 + y = 120
2y = 134
Y = 67

Then
X = 81 – 67 = 14
Z = 120-67 = 53

Answer: 67,14,53