-
Q01 |
- Diagram : Nine point circle
- Q02 | - Diagram : Nine point circle
- Q03 | - Diagram : Nine point circle
- Q04 | - Conclusion
- Q02 | - Diagram : Nine point circle
Q01. Diagram : Nine point circle
|
- Three mid points D, E, F of triangle ABC
- Three feet I, J, K of the heights of triangle ABC
- Three mid points from vertices A, B, C to ortho-center O
- Mid point P of AO
- Mid point Q of BO
- Mid point R of CO
- Prove FP is parallel to BO (or BJ)
- Since F and P are mid-point of triangle ABO
- Prove angle PFD = 90 degrees
- Right angle triangle ABJ : Angle ABJ + angle BAJ = 90 degrees
- Since PF parallel to BO and DF parallel to AC
- Hence angle AFP = angle ABJ and angle BFD = ???
- Hence angle PFD = 90 degrees
- Similarly : Angle PED = 90 degrees
- Hence points D, E, F, P are concuclic and PD is diameter
- Since angle PID = 90 degrees
- Hence I is also on this circle
- Hence D, E, F, I, P are on same circle
Go to Begin
Q02. Diagram : Nine point circle
|
- Similar to Q01 we have
- QE is diameter
- Angle QFE = 90 and angle QDE = 90 degrees
Go to Begin
Q03. Diagram : Nine point circle
|
- Similar to Q01 we have
- RF is diameter
- Angle RKF = 90 and angle RQF = 90 degrees
Go to Begin
Q04. Conclusion
What are the nine points ?
- Three mid points D, E, F of triangle ABC
- Three feet I, J, K of the heights of triangle ABC
- Three mid points from vertices A, B, C to ortho-center O
- Mid point P of AO
- Mid point Q of BO
- Mid point R of CO
- Q01 : D, E, F, I, P on same circle with PD as diameter
- Q02 : D, E, F, J, Q on same circle with QE as diameter
- Q03 : D, E, F, K, R on same circle with RF as diameter
- Hence D, E, F, I, J, K, P, Q, R are same circle
Go to Begin