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 Take Two Incircles, Get Cyclic Quadrilateral? (Posted on 2007-07-28)
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 Comment 1 of 1
`Let I and M be the incenters of triangles PQR and QJRrespectively. For the incircle of triangle QJR to touchQR at point S, incenter M must lie on line segment SI.`
`Circles (Q,|QS|) and (R,|RS|) are tangent at point Sand incircles (I,|IS|) and (M,|MS|) are tangent at point Swith 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 fourcircles  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 theinversions 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 doesnot pass through point S.`
`Therefore, the points T, U, L, and K lie on a circle which isthe 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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