Question

From the given figure, prove that : AP + BQ + CR = BP + CQ + AR. 

Also show that : AP + BQ + CR = `1/2` × Perimeter of ΔABC.

Answer 1

Since from B, BQ and BP are the tangents to the circle

Therefore, BQ = BP   ...(i)

Similarly, we can prove that

AP = AR  ...(ii)

And CR = CQ  ...(iii)

Adding,

AP + BQ + CR = BP + CQ + AR  ...(iv)

Adding AP + BQ + CR to both sides

2(AP + BQ + CR) = AP + PQ + CQ + QB + AR + CR

2(AP + BQ + CR) = AB + BC + CA

Therefore, AP + BQ + CR = `1/2` × (AB + BC + CA)

AP + BQ + CR = `1/2` × perimeter of triangle ABC