Problem 3

Let

P_1,P_2,P_3,P_4

be four points on a circle. For each

i

, let

I_i

be the incenter of the triangle formed by the other three points. Prove that

I_1,I_2,I_3,I_4

are the vertices of a rectangle.

Step 5 of 5: Identify the four vertices with the four incenters

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Q_1=I_4,\qquad Q_2=I_3,\qquad Q_3=I_2,\qquad Q_4=I_1

Detailed analysis

At $Q_1$ , the two circle centers give the angle bisectors of the angles at $P_1,P_2,P_3$ in triangle $P_1P_2P_3$ ; equivalently $Q_1$ lies on all three internal angle bisectors. Thus $Q_1=I_4$ . Cyclically, $Q_2=I_3$ , $Q_3=I_2$ , and $Q_4=I_1$ . Therefore the given four incenters are the vertices of the rectangle just proved.