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Symbol Defintion
Example : Sqr(x) = square root of x
GE 04 01. Circle : Related angles
Diagrams
- Inscribed angle = half of measurement of arc
- Central angle = mesurement of arc
- Tangent and chord angle = inscribed angle of same arc
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GE 04 02. Circles : tangent
Diagrams : Point at outside of circle
- Tangent is perpendicular to the radius at point on circle
- Tangent from outside circle : Two tangents are same length
- Tangent to point on circle
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GE 04 03. Circle : Three points and circle
Diagrams
- Draw circle passing three griven points : Program 04 01
- This circle is called ex-circle (see five centers in triangle)
- 1. Make a triangle using 3 points A, B, C
- 2. Bisect the sides and the bisectors meet at one point E (Ex-center)
- 3. Use distance from ex-center E to vertex A as radius making circle
- 4. Since EA = EB = EC, hence the circle passes A, B, C.
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Study | Program 10 03
- Two points A and B are fixed
- One moving point C to keep angle ACB as constant
- What is the locus of ex-center ?
- It is a point
- Let the ex-center be E
- Since A, B, C on same circle
- Since ACB is constant, hence the ex-center will be the same
- Point E will not move as C moves
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GE 04 04. Circle : Tangent three sides of a triangle
Diagrams
- Definition of in-circle
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Study | Program 10 02
- Two points A and B are fixed
- One moving point C to keep angle ACB as constant
- What is radius ?
- It is an arc
- Let the in-center be I
- The angle AIB is constant if ACB is constant
- Angle IAB = A/2 and angle IBA = B/2
- Hence angle AIB = pi - A/2 - B/2 = pi/2 + C/2
- A and B are fixed and AIB is constant, hence locus of I is arc AIB
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GE 04 05. Five points on circle, How many chords can be drawn
Diagrams : Five points on circle
- If 3 points on circle, how many chords can be made ?
- If 4 points on circle, how many chords can be made ?
- If 11 points on circle, how many chords can be made ?
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GE 04 06. Circle tangents 4 sides of quadrilateral
Diagrams
- Let circle tangents 4 quadrilertal ABCD at E,F,G,H
- Prove that AB + CD = AC + BC
- AF = AE (2 tnagents are equal)
- BF = BG
- CH = HD
- HD = DE
- Hence AB + CD = AC + BC
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GE 04 07. Four point on circle
Diagrams
- Let four points be A,B,C,D
- The angle A + angle C = angle B + angle D
- Angle ACB = (arc AB)/2
- Angle ADB = (arc AB)/2
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GE 04 08.Circle : Tangent three sides of a triangle
Diagrams
- See keyword ex-circle of triangle
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Study | Program 10 03
- Two points A and B are fixed
- One moving point C to keep angle ACB as constant
- What is radius ?
- It is an arc
- Let the ex-center be I
- The angle AIB is constant if ACB is constant
- Angle IAB = A/2 and angle IBA = B/2
- Hence angle AIB = pi - A/2 - B/2 = pi/2 + C/2
- A and B are fixed and AIB is constant, hence locus of I is arc AIB
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GE 04 09. Answer
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GE 04 10. Answer
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GE 04 00. Outlines
Triangle : 3 sides tangent a circle
- Circle is in-circle of triangle
- Tangent from A to incircle is (s - a) where s = (a + b + c)/2
- Radius of incircle = (s - a)*tan(A/2)
- Center : h = s and k = r related with AB
- Circle is ex-circle of triangle
- Tangent from A to incircle is s where s = (a + b + c)/2
- Radius of incircle r = s*tan(A/2)
- Center : h = s and k = r related with AB
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