Question 24523: Somebody please help me with this problem... and thank you in advance!!
USING DEDUCTIVE REASONING
There are 9 significant points associated with a triangle that always lie on a circle. These points are:
- the midpoints of the sides: D, E, F
- the feet of the altitudes: P, Q, R
- the mipoints of the segments joining the orthocentre, H, to the three vertices: X, Y, Z
a) Prove that quadrilaterals FEZY and DEXY are rectangles
b) Expain why the result of part a) proves that D, E, F, X, Y, and Z all lie on the same circle.
c) Why do the points P, Q, and R also lie on this circle?
d) Complete a proof that D, E, F, P, Q, R, X, Y, and Z lie on a circle. This circle is called the nin-point circle of the triangle.
Answer by venugopalramana(3286)   (Show Source):
You can put this solution on YOUR website! USING DEDUCTIVE REASONING
There are 9 significant points associated with a triangle that always lie on a circle. These points are:
- the midpoints of the sides: D, E, F
- the feet of the altitudes: P, Q, R
- the mipoints of the segments joining the orthocentre, H, to the three vertices: X, Y, Z
a) Prove that quadrilaterals FEZY and DEXY are rectangles
b) Expain why the result of part a) proves that D, E, F, X, Y, and Z all lie on the same circle.
c) Why do the points P, Q, and R also lie on this circle?
d) Complete a proof that D, E, F, P, Q, R, X, Y, and Z lie on a circle. This circle is called the nin-point circle of the triangle.
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NOW THAT YOU HAVE NAMED ALL THE POINTS IN THE DIAGRAM,WE SHALL USE THOSE AND ASSUME YOU HAVE A DRAWING NAMED ACCORDINGLY IN FRONT OF YOU.
a) Prove that quadrilaterals FEZY.WE HAVE
IN TRIANGLE ABC,F AND E ARE MID POINTS OF AB AND AC.HENCE FE IS PARALLEL TO BC AND FE=BC/2
IN TRIANGLE HBC,Y AND Z ARE MID POINTS OF HB AND HC.HENCE YZ IS PARALLEL TO BC AND YZ=BC/2
SO FE=YZ.
SIMILARLY
IN TRIANGLE ABH,F AND Y ARE MID POINTS OF AB AND BH.HENCE FY IS PARALLEL TO AH AND FY=AH/2
IN TRIANGLE AHC,E AND Z ARE MID POINTS OF AC AND HC.HENCE EZ IS PARALLEL TO AH AND EZ=AH/2
SO FY=EZ.
FURTHER AS WE PROVED ABOVE FY IS PARALLEL TO AH OR AHP AND YZ IS PARALLEL TO BC.BUT AHP IS PERPENDICULAR TO BC.HENCE FY IS PERPENDICULAR TO YZ.
SIMILARLY WE CAN PROVE THAT FE IS PERPENDICULAR TO EZ.
SO FEZY IS A RECTANGLE.
SIMILARLY WE CAN PROVE THAT
DEXY IS A RECTANGLE.
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b) Expain why the result of part a) proves that D, E, F, X, Y, and Z all lie on the same circle.
WE PROVED ABOVE THAT
quadrilaterals FEZY and DEXY are rectangles.FURTHER THEY HAVE A COMMON DIAGONAL EY.WE KNOW IN RECTANGLES DIAGONALS ARE EQUAL AND BISECT EACH OTHER.SO IN THESE 2 RECTANGLES DIAGONALS FZ,EY AND DX ARE ALL EQUAL AND BISECT EACH OTHE.LET THE POINT OF BISECTION BE N .SO IF WE DRAW A CIRCLE WITH N AS CENTRE AND ANY OF THE 3 DIAGONAL LENGTHS AS DIAMETER THEN ,SINCE ALL THE 3 DIAGONALS ARE EQUAL IN LEMGTH , THAT CIRCLE WILL PASS THROUGH AL THE ENDS OF THE 3 DIAGONALS.HENCE F,E,Z,Y,D AND X ARE CONCYCLIC OR LIE ON THE SAME CIRCLE.
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c) Why do the points P, Q, and R also lie on this circle?
DX IS A DIAMETER OF THE ABOVE CIRCLE.WE KNOW THAT ANGLE DPX IS 90 DEGREES AS AP IS SAME AS BDPC IS PERPENDICULAR TO AP OR AXP.SINCE ANGLE IN A SEMICIRCLE IS 90 DEGREES . HENCE A CIRCLE WITH DX AS DIAMETER WILL PASS THROUGH P.
SIMILARLY WE CAN PROVE THAT Q AND R ALSO LIE ON THE SAME CIRCLE
d) Complete a proof that D, E, F, P, Q, R, X, Y, and Z lie on a circle. This circle is called the nin-point circle of the triangle.
AS PROVED ABOVE ALL 9 POINTS D, E, F, P, Q, R, X, Y, and Z lie on a circle
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