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 3 of 5: Use angle chasing to obtain the rectangle directions

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Q_1Q_4\parallel Q_2Q_3,\qquad Q_1Q_2\parallel Q_3Q_4,\qquad Q_1Q_4\perp Q_1Q_2

Detailed analysis

The angle equalities in the auxiliary circles give $Q_1Q_4\parallel Q_2Q_3$ and $Q_1Q_2\parallel Q_3Q_4$ . The centers of successive auxiliary circles are arc midpoints on $C$ , and the corresponding center chords are perpendicular; hence $Q_1Q_4\perp Q_1Q_2$ .