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Orthogonal Circle (Posted on 2008-05-25) Difficulty: 3 of 5
Let three circles be located such that their centers are non-collinear.

Construct a fourth circle that is orthogonal to each of the three circles.

  Submitted by Bractals    
Rating: 4.0000 (1 votes)
Solution: (Hide)
CONSTRUCTION:

Let circle C4 intersect circles C1, C2, and C3 at
points P41, Q41, P42, Q42, P43, and Q43 respectively.
Let circle C5 intersect circles C1, C2, and C3 at
points P51, Q51, P52, Q52, P53, and Q53 respectively.

Let

   X412 = P41Q41 /\ P42Q42
   X512 = P51Q51 /\ P52Q52
   X423 = P42Q42 /\ P43Q43
   X523 = P52Q52 /\ P53Q53

   R = X412X512 /\ X423X523
where /\ denotes intersection.

The desired circle is a circle with center R and orthogonal to any one of the circles C1, C2, and C3.

PROOF:

PxyQxy is the radical axis of circles Cx and Cy.

Xxyz is a point on the radical axis of circles Cy and Cz.

X4yzX5yz is the radical axis of circles Cy and Cz.

R is the radical center of circles C1, C2, and C3.

A circle with center R and orthogonal to any one of the
circles C1, C2, and C3 is orthogonal to all three and is called the radical circle.

Comments: ( You must be logged in to post comments.)
  Subject Author Date
re: What is geometric construction?Charlie2008-05-26 12:56:51
What is geometric construction?Dej Mar2008-05-26 10:53:45
re: An answerBractals2008-05-25 15:54:16
QuestionAn answerDej Mar2008-05-25 13:02:15
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