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Symbol Defintion
Example : x^2 = square of x
GE 19 01. Diagram of nine point circle
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GE 19 02. The nine points of triangle
The feet of three altidues
- I : AI is altitude
- J : BJ is altitude
- K : CK is altitude
- D : AD is median
- E : BE is median
- F : CF is median
- P is mid point of AO
- Q is mid point of BO
- R is mid point of CO
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GE 19 03. The diameters of nine points circle
Diameters
- PD
- QE
- RE
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GE 19 04. Proof of P, D, E, F are concyclic
Triangle ABO
- Since P and F are mid pointc of triangle, hence PF is parallel to BO
- Hence angle AFP = angle ABJ (Corresponding angles)
- Since D and E are mid points of triangle, hence DF is parallel to AC
- Hence angle BFD = angle BAC (Corresponding angles)
- angle ABJ + angle BAC = 90 degrees
- Hence angle AFP + angle BFD = 90
- Hence angle PFD = 90 (Since angles AFP + PFD + BFD = 180)
- Similarly we can prove that angle PED = 90 degrees
- Angles PFD and PED are opposite angles and their sum is 180 degrees
- Hence, P, D, E, F are concylic
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GE 19 05. Prove that Q,D,E,F are concyclic
Proof
- Similar to GE 19 04
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GE 19 06. Prove that R,D,E,F are concyclic
Proof
- Similar to GE 19 04
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GE 19 07. Conclusion
D,E,F,P,Q,R are same circle
- From Q04, Q05, Q06 we get the conclusion
- Right triangle PDI and PD is diameter of the circle, I is on circle
- Right triangle QEJ and QE is diameter of the circle, J is on circle
- Right triangle RFK and RF is diameter of the circle, K is on circle
- Since PD, QE abd RF are diameters of the same circle
- Hence D,E,F,I,J,K,P,Q,R are on same circle
- The proof is from the nine points circle.
- The proof can be done using construction and measurement
- The proof can also be done using coordinate geometry
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GE 19 08. Answer
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GE 19 09. Answer
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GE 19 10. Diagrams in geometry
Constructions
- How to bisect a line ? Program 02 01
- How to bisect an angle ? Program 02 02
- How to draw a circle to tongnet three sides of a triangle ? Program 03 02
- How to draw a circle to pass three given points ? Program 03 02, 04 01
- How to draw a pedal triangle ? Program 03 03
- How to draw a ex-central triangle ? Program 03 04
- How to draw tangents to circle from point outside of circle ? Program 04 02
- How to draw tangent to circle at point on the circle ? Program 04 03
- 1. What is es-circle of a triangle ? Program 03 02
- 2. What is ex-circle of a triangle ? Program 03 02
- 3. What is in-circle of a triangle ? Program 03 02
- 4. What is gravity-center of a triangle ? Program 03 02
- 5. What is ortho-center of a triangle ? Program 03 02
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GE 19 00. Outlines
Nine points circle
- Three points are the mid-points of triangle
- Three points are the feet of altitudes
- Three points are the mid-points from vertices of triangle to ortho-center
- The theory of mid-points of triangle
- Four points P, D, E, F are concylic and then angle P + E = angle D + F = 180 degrees
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