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Problem 3

Let P1,P2,P3,P4P_1,P_2,P_3,P_4 be four points on a circle. For each ii, let IiI_i be the incenter of the triangle formed by the other three points. Prove that I1,I2,I3,I4I_1,I_2,I_3,I_4 are the vertices of a rectangle.
Step 1 of 5: Construct four auxiliary circles
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C12,C23,C34,C41 have centers O12,O23,O34,O41 on CC_{12},C_{23},C_{34},C_{41}\text{ have centers }O_{12},O_{23},O_{34},O_{41}\text{ on }C
Detailed analysis

For each consecutive pair (Pi,Pi+1)(P_i,P_{i+1}), draw a circle Ci,i+1C_{i,i+1} whose center Oi,i+1O_{i,i+1} lies on the given circle CC and which passes through that pair. Indices are cyclic, so P5=P1P_5=P_1.