perplexus.info :: Geometry : Locus problem

Locus problem (Posted on 2007-05-27)

Let point D lie on side BC of triangle ABC.
Let C₁ and C₂ be the incircles of triangles ABD and ACD respectively.
Let m be the common external tangent to C₁ and C₂ different from BC.
Let P be the intersection of line AD with m.

What is the locus of point P as point D varies between B and C?

Submitted by Bractals

Rating: 4.0000 (1 votes)

Solution: (Hide)

Let E, F, G, and H be the tangent points of C₁ with AC, BC, AD, and m respectively.
Let K, L, M, and N be the tangent points of C₂ with AD, BC, AD, and m respectively.
2|AP| = |AP| + |AP| = (|AG| - |PG|) + (|AM| - |PM|)

= (|AE| - |PH|) + (|AK| - |PN|)

= (|AE| - |PH|) + (|EB| - |BF|) + (|AK| - |PN|) + (|KC| - |LC|)

= (|AE| + |EB|) + (|AK + |KC|) - |BF| - (|HP| + |PN|) - |LC|

= |AB| + |AC| - (|BF| + |HN| + |LC|)

= |AB| + |AC| - (|BF| + |FL| + |LC|)

= |AB| + |AC| - |BC|
Therefore, the locus of point P is a circular arc with center A and
radius (|AB| + |AC| - |BC|)/2.

Comments: ( You must be logged in to post comments.)

	Subject	Author	Date
	Unproven solution	brianjn	2007-06-01 05:04:41

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