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Take Two Incircles, Get Cyclic Quadrilateral? (Posted on 2007-07-28) Difficulty: 3 of 5
Consider a triangle PQR whose incircle touches the sides QR, RP and PQ respectively at the points S, T and U.

J is a point which is located inside the triangle PQR such that the incircle of the triangle QJR touches QR, RJ and JQ respectively at the points S, K and L.

Determine whether or not the quadrilateral TULK is cyclic.

See The Solution Submitted by K Sengupta    
Rating: 3.0000 (1 votes)

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Solution Solution Comment 1 of 1

Let I and M be the incenters of triangles PQR and QJR
respectively. For the incircle of triangle QJR to touch
QR at point S, incenter M must lie on line segment SI.
Circles (Q,|QS|) and (R,|RS|) are tangent at point S
and incircles (I,|IS|) and (M,|MS|) are tangent at point S
with SI perpendicular to QR.
The intersections of these four circles (excluding point S)
are the points T, U, L, and K.
Inversion through a circle with center S inverts these four
circles  into four lines Q', R', I', and M' with Q' and R'
perpendicular to QR and I' and M' parallel to QR.
These four lines determine a rectangle whose vertices are the
inversions of points T, U, L, and K.
A circle W can be easily be circumscribed about this rectangle.
Since vertex J is located inside triangle PQR, circle W does
not pass through point S.
Therefore, the points T, U, L, and K lie on a circle which is
the inversion of circle W.

Edited on July 28, 2007, 10:14 pm
  Posted by Bractals on 2007-07-28 22:09:13

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