perplexus.info :: Geometry : Locus of Points

Locus of Points (Posted on 2007-12-12)

Let X₁, X₂, ... , X_n be n≥2 distinct points on a circle C with center O and radius r. What is the locus of points P inside C such that

∑ⁿ_i=1 |X_iP|/|PY_i| = n

,where the line X_iP also intersects C at point Y_i.

Submitted by Bractals

Rating: 3.0000 (2 votes)

Solution: (Hide)

Clearly the center O is in the locus. Let point P ≠ O be inside C.
Let the line OP intersect C at points A and B with P between O and B.
∑ⁿ_i=1 |X_iP|/|PY_i| = ∑ⁿ_i=1 |X_iP|²/|X_iP|·|PY_i| = ∑ⁿ_i=1 X_iP•X_iP/|AP|·|PB| = ∑ⁿ_i=1 (OP - OX_i)•(OP - OX_i)/(r + |OP|)·(r - |OP|) = [∑ⁿ_i=1 |OP|² + |OX_i|² - 2OP•OX_i] / [r² - |OP|²] = [n(r² + |OP|²) - 2OP•∑ⁿ_i=1 OX_i] / [r² - |OP|²] = [n(r² + |OP|²) - 2OP•nOG] / [r² - |OP|²] = n[r² + (|OP|² - 2OP•OG)] / [r² - |OP|²] = n if OP•OG = |OP|².
,where G is the center of mass of the points X₁, X₂, ... , X_n.
Clearly the center of mass G is in the locus since OG•OG = |OG|².
OP•OG = |OP|² <==> OP•OG - |OP|² = 0 <==> OP•(OG - OP) = 0 <==> OP•PG = 0
But, the locus of points P satisfying OP•PG = 0 is the circle with OG as a diameter.

Note: CD denotes the vector from point C to point D.

Note: CD•EF denotes the dot product (scaler product) of vectors CD and EF.

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

Subject	Author	Date
Puzzle Answer	K Sengupta	2023-09-30 07:59:20
re: Solution	Charlie	2007-12-15 18:13:44
re: Solution	Charlie	2007-12-14 12:52:55
re: Solution	Charlie	2007-12-14 10:09:42
Solution	Praneeth	2007-12-14 08:25:37
re(3): Solution (spoiler)	Charlie	2007-12-14 00:47:20
re(2): Solution (spoiler)	Praneeth	2007-12-13 23:11:57
re: Solution (spoiler)	Charlie	2007-12-13 10:34:15
Solution (spoiler)	Praneeth	2007-12-13 01:44:59
and also	Charlie	2007-12-13 00:21:20
re: three points again -- that second one got cut off	Charlie	2007-12-13 00:19:33
three points again	Charlie	2007-12-13 00:18:04
Hint	Praneeth	2007-12-13 00:07:18
three regularly spaced points	Charlie	2007-12-12 22:06:32
third point	Charlie	2007-12-12 22:04:34
re: exploration with computer -- 2 non-opposing points	Charlie	2007-12-12 19:52:15
exploration with computer	Charlie	2007-12-12 19:48:10

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