Online Geometry theorems, problems, solutions, and related topics.
See complete Problem 70Squares Inscribed in a Triangle. Level: High School, SAT Prep, College geometryPost your solutions or ideas in the comments.
Proof.If n = 1 then s↓1= bh/(b+h), as showed in the problem proposed 69 (with proof posted in September 9, 2009). If n = 2 we have s↓2= s↓1h↓2/(s↓1+h↓2), where h↓2 is the height of the triangle with vertex B and base s↓1.From h↓2= h - s↓1 we have h↓2= h - bh/(b+h) = h↑2./(b+h). So,s↓2= s↓1h↓2/(s↓1+h↓2) = (bh/(b+h))(h↑2./(b+h))/((bh/(b+h)+(h↑2./(b+h)) = bh↑2./(b+h)↑2.Therefore, by induction we have s↓n= bh↑n./(b+h)↑n.QED, Ianuarius.