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 4 of 5: Conclude that the second-intersection quadrilateral is a rectangle

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Q_1Q_2Q_3Q_4\text{ is a rectangle}

Detailed analysis

The parallel relations from the preceding step give both pairs of opposite sides of $Q_1Q_2Q_3Q_4$ parallel, so it is a parallelogram. One pair of adjacent sides is perpendicular, hence it is a rectangle.