1. **Stating the problem:** Given a triangle ABC inscribed in a circle with center O, points H, N, and M lie on or inside the triangle and circle. Lines AH, BN, and CM are drawn from vertices. Right angles are marked at H on BH and at N on BN. We need to analyze the geometric relationships and possibly prove properties related to these points and lines. 2. **Key concepts and formulas:** - In a circle, an inscribed angle subtends a chord. - Right angles in a circle often indicate that points lie on a circle with diameter as the hypotenuse (Thales' theorem). - Perpendicular lines from vertices to opposite sides or chords can indicate altitudes or orthocenter properties. 3. **Analyzing the right angles:** - Since \(\angle BHN = 90^\circ\), point H lies on the circle with diameter BN. - Since \(\angle BNM = 90^\circ\), point N lies on the circle with diameter BM. 4. **Using Thales' theorem:** - For \(\angle BHN = 90^\circ\), H lies on the circle with diameter BN. - For \(\angle BNM = 90^\circ\), N lies on the circle with diameter BM. 5. **Conclusion:** - Points H and N lie on circles defined by diameters BN and BM respectively. - This implies that lines AH, BN, and CM are altitudes or related to the orthocenter of triangle ABC. Final answer: Points H and N lie on circles with diameters BN and BM respectively, confirming the right angles and perpendicularity in the triangle inscribed in the circle with center O.