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Four Congruent Circles (Posted on 2006-06-14) Difficulty: 3 of 5
Show how to construct four congruent circles, inside an acute triangle ABC, with centers A', B', C', and D' such that
  1. circle with center A' is tangent to sides AB and AC,
  2. circle with center B' is tangent to sides BC and BA,
  3. circle with center C' is tangent to sides CA and CB, and
  4. circle with center D' is externally tangent to the other three circles.

See The Solution Submitted by Bractals    
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Solution If this is what is meant then this is a solution | Comment 1 of 5

The way this puzzle is phrased might be taken to imply tht the triangle ABC is given, and the circles are to be constructed from that given triangle.

But the puzzle doesn't literally say the triangle is a given, so we could interpret it as asking to construct an entire such system of circles and triangle.  That makes it easier.

Start with the circle centered on D'. Construct it in any size you choose that you'd like all the circles to have.

Construct three rays from D' such that they intersect the circle in three points not all within any 180-degree arc (otherwise you'd get a right or obtuse triangle eventually).

Mark off one radius distance along each ray from the points of intersection of the rays with the circle, away from the centers of the circles. Label the points thus marked off, A', B' and C'.  Then construct new circles centered on each of A', B' and C' with the same radius as before.  Now you have all the circles externally tangent to the one centered on D'.

Connect A' to B', B' to C' and C' to A'.  Construct perpendiculars to the endpoints of each of these segments. The intersection of each of these perpendiculars with the circles, on the side external to the construction so far, will give two points on each of the sides of the triangle to be formed as they are tangent to two circles each.  Label the intersections of these lines A, B and C, paying attention to which circles' centers are so labeled, making sure they correspond.

The entire construction is then finished.

If you're supposed to do this when triangle ABC is already given, I don't know how.

  Posted by Charlie on 2006-06-14 15:50:31
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