Problem 5

Three congruent circles have a common point

O

and lie inside a given triangle. Each circle is tangent to two of the triangle's sides. Prove that the incenter and the circumcenter of the triangle, together with the point

O

, are collinear.

Step 3 of 6: Show corresponding sides are parallel

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O_BO_C \parallel BC,\quad O_CO_A \parallel CA,\quad O_AO_B \parallel AB

Detailed analysis

Because the circles are congruent, with common radius $\rho$ , and $O_B,O_C$ are both at distance $\rho$ from line $BC$ , segment $O_BO_C$ is parallel to $BC$ ; the same argument applied to the other two sides gives the remaining two parallels.