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Question
From the given figure, prove that : AP + BQ + CR = BP + CQ + AR.
Also show that : AP + BQ + CR = `1/2` × Perimeter of ΔABC.
Solution
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
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