See complete Problem 70

Squares Inscribed in a Triangle. Level: High School, SAT Prep, College geometry

Post your solutions or ideas in the comments.

## Monday, May 19, 2008

### Elearn Geometry Problem 70

Labels:
similarity,
square,
triangle

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Proof.

ReplyDeleteIf 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.