1. **Problem Statement:** We are given a triangle ABC with an inscribed circle (incircle) touching the sides at points P, Q, and R. The circle has center O and radius $r$. We need to analyze the given expressions: i) $AB + CB = AC + BQ$ ii) The area of triangle $ABC$ is given by $$\text{Area} = \frac{1}{2} \times \text{perimeter} \times r$$ 2. **Understanding the incircle and tangent segments:** The incircle touches each side of the triangle at exactly one point, creating tangent segments from each vertex to the points of tangency. These tangent segments from a vertex to the points of tangency on the two adjacent sides are equal in length. Let the tangent lengths from vertices be: - From vertex $A$: $AP = AR = x$ - From vertex $B$: $BP = BQ = y$ - From vertex $C$: $CQ = CR = z$ 3. **Expressing side lengths in terms of tangent lengths:** - Side $AB = AP + PB = x + y$ - Side $BC = BQ + QC = y + z$ - Side $AC = AR + RC = x + z$ 4. **Verifying the given equation i):** Given: $AB + CB = AC + BQ$ Substitute: $$AB + CB = (x + y) + (y + z) = x + 2y + z$$ $$AC + BQ = (x + z) + y = x + y + z$$ Since $x + 2y + z \neq x + y + z$ in general, the given equation as stated is not correct unless $y=0$ which is impossible for a triangle. Possibly the problem meant $AB + BC = AC + 2BQ$ or a similar relation involving tangent lengths. 5. **Formula for area using inradius:** The area $A$ of triangle $ABC$ can be expressed as: $$A = r \times s$$ where $r$ is the inradius and $s$ is the semiperimeter: $$s = \frac{AB + BC + AC}{2}$$ The problem states: $$\text{Area} = \frac{1}{2} \times \text{perimeter} \times r = r \times s$$ which is correct because perimeter $= 2s$. 6. **Summary:** - Tangent segments from each vertex to the incircle are equal. - Side lengths can be expressed as sums of tangent lengths. - Area of the triangle equals inradius times semiperimeter. **Final answers:** - The given equation i) is not generally true as stated. - The area formula ii) is correct: $$\text{Area} = r \times s = \frac{1}{2} \times \text{perimeter} \times r$$