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Q01 |
- Diagram : Diagram of Butterfly Theorem
- Q02 | - Definition of Butterfly Theorem
- Q03 | - Proof of Butterfly Theorem
- Q04 | - Second step to prove
- Q05 | - Theorems used in proof
- Q02 | - Definition of Butterfly Theorem
Q01. Diagram : Hyperbola x*y = 1
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Q02. Definition of butterfly theorem
Draw diagram
- Draw a circle
- Draw chord AB and chord CD. Two chords intersect at M
- Draw line PQ and passing M
- Join AD and BC
- Let AD cut PQ at X and BC cut PQ at Y
- If PM = MQ, Then XM = MY
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Q03. Proof of buttterfly theorem
Construction
- Draw line XU perpendicular to AB. Let XU = x1
- Draw line XV perpendicular to CD. Let XV = x2
- Draw line YS perpendicular to AB. Let YS = y1
- Draw line YT perpendicular to CD. Let YT = y2
- Let XM = x and YM = y
- Triangle XUM similar to YMS
- Angle XMU = angle YMS
- Triangle XMU and triangle YMS are right angle triangle
- Hence these two triangle are similar
- Hence x/y = x1/y1
- Similarly, x/y = x2/y2
- Hence x/y = x1/y1 = x2/y2
- Triangle AXU is similar to triangle CYT
- They are right angle triangle
- Angle A = angle C (Face same arc BD)
- Hence x1/y2 = AX/CY
- Triangle XVD is similar to triangle YSB
- They are right angle triangle
- Angle B = angle D (Face same arc AC)
- Hence x1/y2 = AX/CY
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Q04. Proof XM = YM
From (1), (2) and (3)
- From (1) we have x/y = x1/y1 = x2/y2
- (x/y)^2 = (x1*x2)/(y1*y2)
- = (x1/y2)/(x2/y1)
- = (AX/CY)/(XD/YB)
- = (AX*XD)/(CY*YB) ...................(4)
- Two chrod theorem :
- Chords AD and PQ : AX*XD = PX*XQ
- Chords CD and PQ : CY*YB = PY*YQ
- Hence (4) becomes
- (x/y)^2 = (PX*XD)/(PY*YQ) ..... (5)
- Assume that PM = a and MQ = a
- Hence PX = PM - XM = a - x
- Hence XD = XM + MQ = a + x
- Hence PY = PM + MY = a + y
- Hence YQ = MQ - YM = a - y
- Hence (5) becomes
- (x/y)^2 = ((a - x)*(a + x))/((a + y)*(a - y))
- (x/y)^2 = (a^2 - x^2)/(a^2 - y^2)
- Hence (x^2)*(a^2 - y^2) = (y^2)*(a^2 - x^2)
- Hence (x^2)/(y^2) = 1
- Hence x = y and XM = MY
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Q05. Theorem used
Similar of two triangle
- Triangle ABC is simialer to triangle A'B'C'
- Then AB/A'B' = BC/B'C' = CD/C'D'
- Chords AB and CD intersect at P
- Then AP*PB = CP*PD
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