perplexus.info :: Geometry : Kissing Incircles

Kissing Incircles (Posted on 2007-06-01)

Let point D lie on side BC of triangle ABC.
Let C1 and C2 be the incircles of triangles ABD and ACD respectively.

If C1 and C2 are tangent, then write |BD|/|DC| in terms of d;
where d = (|CA| - |AB|)/|BC|.

Submitted by Bractals

Rating: 2.5000 (2 votes)

Solution: (Hide)

Let r = |BD|/|DC|. Then |BD| = r|BC|/(1+r) and |DC| = |BC|/(1+r).

Let D₁ be the point of tangency between C₁ and AD.
Let D₂ be the point of tangency between C₂ and AD.

The incircles will kiss when
|DD₁| = |DD₂| |AD| + |BD| - |AB| |AD| + |DC| - |CA| -------------------- = -------------------- 2 2 |BD| - |DC| = |AB| - |CA| 1 - r |CA| - |AB| ------- = ------------- = d 1 + r |BC| 1 - d r = ------- 1 + d

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