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 2 of 5: Define the four second intersections

Previous step Next step

Q_1=C_{12}\cap C_{23}\setminus\{P_2\},\quad Q_2=C_{23}\cap C_{34}\setminus\{P_3\},\quad Q_3=C_{34}\cap C_{41}\setminus\{P_4\},\quad Q_4=C_{41}\cap C_{12}\setminus\{P_1\}

Detailed analysis

Let $Q_1,Q_2,Q_3,Q_4$ be the second intersections of consecutive auxiliary circles as displayed. The centers bisect the relevant arcs of $C$ , so the radii and inscribed-angle relations determine the directions of the lines joining the $Q_i$ .