Geometry Problem. Post your solution in the comment box below.

Level: Mathematics Education, High School, Honors Geometry, College.

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## Tuesday, March 10, 2015

### Problem 1095: Externally Tangent Circles, Common Tangent, Radius, Geometric Mean, Metric Relations

Labels:
common tangent,
geometric mean,
metric relations,
radius,
tangent

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d1 / r1 = d2 / r2 = (d1 - d2) / (r1 - r2)

ReplyDelete={sqrt[(r1 + r2)^2 - (r1 - r2)^2]} / (r1 - r2)

= 2sqrt(r1.r2) / (r1 - r2)

from which the result follows.

Let x=O₂P.

ReplyDeleter₁/r₂ = (r₁+r₂+x)/x = 1 + (r₁+r₂)/x

x = r₂(r₁+r₂)/(r₁-r₂)

d₂² = x² - r₂²

= r₂² (4r₁r₂)/(r₁-r₂)²

d₂ = 2r₂ √(r₁r₂) / (r₁ - r₂)

d₁/d₂ = r₁/r₂

d₁ = 2r₁ √(r₁r₂) / (r₁ - r₂)

A₁A₂² =(r₁+r₂)²- (r₁-r₂)²=4r₁r₂=>d₁-d₂= 2 √(r₁r₂)

ReplyDeleted₁/d₂ = r₁/r₂=>d₁/(d₁-d₂)=r₁/(r₁-r₂)=>d₁ = 2r₁ √(r₁r₂) / (r₁ - r₂);d2/(d₁-d₂)=r2/(r₁-r₂)=>d₂ = 2r₂ √(r₁r₂) / (r₁ - r₂)